#6364112

order	512
nilpotency class	3
isoclinism	isoclinic class of #6363930
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#3205	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1485 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2)(3,12)(4,9)(5,10)(6,20)(7,11)(8,15)(13,23)(14,29)(16,24)(17,25)(18,32)(19,27)(21,28)(22,31)(26,30)(33,34,39,43)(35,40,46,55)(36,48,50,56)(37,42,38,51)(41,52,57,49)(44,58,60,63)(45,54,47,61)(53,62,64,59);(1,3,7,16)(2,8,11,23)(4,15,6,13)(5,18,19,31)(9,24,20,12)(10,26,27,29)(14,21,30,17)(22,25,32,28)(33,35)(34,40)(36,44)(37,45)(38,47)(39,46)(41,53)(42,54)(43,55)(48,58)(49,59)(50,60)(51,61)(52,62)(56,63)(57,64);(1,4,7,6)(2,9,11,20)(3,13,16,15)(5,17,19,21)(8,12,23,24)(10,25,27,28)(14,18,30,31)(22,29,32,26)(33,36,39,50)(34,41,43,57)(35,44,46,60)(37,49,38,52)(40,53,55,64)(42,56,51,48)(45,59,47,62)(54,63,61,58);(1,5,7,19)(2,10,11,27)(3,14,16,30)(4,17,6,21)(8,22,23,32)(9,25,20,28)(12,29,24,26)(13,18,15,31)(33,37,39,38)(34,42,43,51)(35,45,46,47)(36,49,50,52)(40,54,55,61)(41,56,57,48)(44,59,60,62)(53,63,64,58);(2,11)(8,23)(9,20)(10,27)(12,24)(22,32)(25,28)(26,29)(35,46)(40,55)(44,60)(45,47)(53,64)(54,61)(58,63)(59,62)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306805, #72306814, #72306833
maximal normal subgroup of	#72306805, #72306814, #72306833

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6271308);