#7658216

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=hi,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, 7565412);