#7681353

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	#43009	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4652 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)(5,7)(6,8)(10,14)(15,16)(17,18,23,20)(19,27,28,24)(21,25,30,31)(22,32,29,26);(1,3,6,5)(2,4,8,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,28)(18,24,20,27)(21,22,30,29)(25,26,31,32);(1,3,6,5)(2,7,8,4)(19,28)(21,30)(24,27)(25,31);(1,4,6,7)(2,3,8,5)(9,11,13,12)(10,15,14,16)(17,20,23,18)(19,27,28,24)(21,31,30,25)(22,32,29,26);(1,3,6,5)(2,4,8,7)(9,12,13,11)(10,16,14,15)(17,21,23,30)(18,25,20,31)(19,29,28,22)(24,32,27,26);(1,5,6,3)(2,7,8,4)(17,22,23,29)(18,26,20,32)(19,30,28,21)(24,31,27,25)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7588549);