#272566

order	512
nilpotency class	3
isoclinism	isoclinic class of #272539
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3482	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

763 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 179762);