#7671200

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#39956	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4761 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)(3,4)(5,8)(6,7)(10,13)(14,16)(17,18)(19,27)(20,25)(21,31)(22,24)(23,28)(26,29)(30,32);(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,27)(25,31)(26,32);(2,7)(3,8)(18,24)(19,28)(20,30)(26,31);(1,3,6,8)(2,4,7,5)(17,20,22,30)(18,25,24,32)(19,29,28,21)(23,31,27,26);(1,4,6,5)(2,3,7,8)(9,11,12,15)(10,14,13,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,31)(30,32);(1,5,6,4)(2,8,7,3)(17,21,22,29)(18,26,24,31)(19,30,28,20)(23,32,27,25)

Polycyclic presentation:

2^9|b^2=gi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=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, 7578396);