#7658541

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

6820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565737);