#7683602

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	#59755	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4665 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)(3,7,5,4)(9,10,13,12)(11,15,16,14)(17,18,22,24)(19,23,27,31)(20,25,26,21)(28,29,30,32);(1,3,6,5)(2,7,8,4)(9,11,13,16)(10,14,12,15)(17,19)(18,23)(20,30)(21,29)(22,27)(24,31)(25,32)(26,28);(2,8)(4,7)(9,10,13,12)(11,14,16,15)(18,24)(19,27)(21,25)(28,30);(1,4)(2,3)(5,8)(6,7)(9,12,13,10)(11,15,16,14)(17,19)(18,23)(20,30)(21,29)(22,27)(24,31)(25,32)(26,28);(17,20,22,26)(18,25,24,21)(19,28,27,30)(23,29,31,32);(1,5,6,3)(2,4,8,7)(17,21,22,25)(18,26,24,20)(19,29,27,32)(23,28,31,30)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7590798);