#270734

order	512
nilpotency class	3
isoclinism	isoclinic class of #270683
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	#62107	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

934 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,6,3,7,9,5,11)(4,12,13,16,14,10,15,8)(17,18)(19,23)(20,25)(21,24)(22,26)(27,32)(28,30)(29,31);(1,3)(2,5)(4,10)(6,9)(7,11)(8,13)(12,14)(15,16)(17,19,21,29)(18,23,24,31)(20,28,22,27)(25,30,26,32);(1,4)(2,8)(3,10)(5,13)(6,15)(7,14)(9,16)(11,12)(18,24)(19,27)(20,22)(23,30)(28,29)(31,32);(1,2,7,9)(3,5,11,6)(4,8,14,16)(10,13,12,15)(17,20,21,22)(18,25,24,26)(19,28,29,27)(23,30,31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 177930);