#7712527

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

6832 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)(8,15)(12,17)(13,19)(18,20)(21,22,27,24)(23,30,29,26)(25,32,33,28)(31,34,35,36);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)(21,23)(22,26)(24,30)(25,31)(27,29)(28,34)(32,36)(33,35);(6,10)(11,16)(12,17)(18,20)(22,24)(23,29)(25,33)(34,36);(5,7,9,14)(6,11,10,16)(8,13,15,19)(12,18,17,20)(21,24,27,22)(23,30,29,26)(25,32,33,28)(31,36,35,34);(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)(21,25,27,33)(22,28,24,32)(23,31,29,35)(26,34,30,36);(5,8,9,15)(6,12,10,17)(7,13,14,19)(11,18,16,20)(21,26,27,30)(22,29,24,23)(25,34,33,36)(28,35,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619723);