#7682531

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	#59819	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=gh,e^2=g,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, 7589727);