#7678321

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	#45417	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4669 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=g,e^2=gi,f^2=g,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, 7585517);