#8001745

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

7343 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

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, 7908941);