#7629615

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5190 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)(9,10,13,14)(11,15,12,16)(17,18,22,20)(19,23,26,27)(21,24,29,28)(25,30,31,32);(2,5)(6,8)(17,19,22,26)(18,23,20,27)(21,25,29,31)(24,30,28,32);(10,14)(15,16)(17,20,22,18)(19,23,26,27)(21,28,29,24)(25,30,31,32);(9,11,13,12)(10,15,14,16)(17,21)(18,24)(19,25)(20,28)(22,29)(23,30)(26,31)(27,32);(9,12,13,11)(10,16,14,15)(21,29)(24,28)(25,31)(30,32);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^2=h,e^2=h,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, 7536811);