#232888

order	512
nilpotency class	3
isoclinism	isoclinic class of #232876
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=ghi,e^c=ei,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140084);