#7741173

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

6905 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)(6,10)(11,17)(12,15)(18,20)(21,22)(23,29)(24,26)(25,28)(27,31)(30,35)(32,33)(34,36);(1,2)(3,4)(5,6)(7,11)(8,15)(9,10)(12,16)(13,17)(14,20)(18,19)(21,22,26,24)(23,27,31,29)(25,32,33,28)(30,35,36,34);(6,10)(7,13)(12,15)(14,19)(22,24)(27,29)(28,32)(34,35);(5,6)(7,11)(8,12)(9,10)(13,17)(14,18)(15,16)(19,20)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35);(5,7)(6,11)(8,14)(9,13)(10,17)(12,18)(15,20)(16,19)(21,24,26,22)(23,29,31,27)(25,32,33,28)(30,35,36,34);(5,8)(6,12)(7,14)(9,16)(10,15)(11,18)(13,19)(17,20)(21,25,26,33)(22,28,24,32)(23,30,31,36)(27,34,29,35)

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=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, 7648369);