#7631957

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	#35564	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5228 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539153);