#7629671

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#35831	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5211 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,c^b=ci,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7536867);