#6383588

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2404	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=fg,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ci,d^b=dfi,e^c=ei,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6290784);