#7632453

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	#36455	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5258 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=i,e^2=i,f^2=h,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, 7539649);