#7658135

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6624 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

(2,4)(5,6,10,8)(7,13,15,11)(9,12,17,16)(14,19,20,18)(21,22)(23,27)(24,31)(25,29)(26,30)(28,34)(32,36)(33,35);(1,2,3,4)(5,6,10,8)(7,11,15,13)(9,12,17,16)(14,18,20,19)(22,23)(28,31)(29,30)(35,36);(5,7,10,15)(6,11,8,13)(9,14,17,20)(12,18,16,19)(21,23,27,22)(24,31,34,28)(25,30,26,29)(32,36,33,35);(21,24,27,34)(22,28,23,31)(25,32,26,33)(29,35,30,36);(5,8,10,6)(7,13,15,11)(9,12,17,16)(14,18,20,19)(21,25,27,26)(22,29,23,30)(24,32,34,33)(28,35,31,36);(5,9,10,17)(6,12,8,16)(7,14,15,20)(11,18,13,19)(21,26,27,25)(22,30,23,29)(24,33,34,32)(28,36,31,35)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565331);