#7766331

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

6861 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)(5,6)(7,15)(8,12)(9,13)(10,14)(11,16)(17,18)(19,20)(22,27)(23,28)(29,35)(31,33);(1,2,3,4)(5,7)(6,11)(8,9)(10,16)(12,20)(13,17)(14,15)(18,19)(21,22)(23,30)(24,28)(25,29)(26,27)(31,36)(32,33)(34,35);(7,16)(9,19)(11,15)(13,20)(21,23,26,28)(22,24,27,30)(25,33,34,31)(29,36,35,32);(5,8)(6,12)(7,9)(10,18)(11,20)(13,15)(14,17)(16,19)(21,24,26,30)(22,28,27,23)(25,32,34,36)(29,33,35,31);(5,6)(7,11)(8,17)(9,13)(10,14)(12,18)(15,16)(19,20)(21,22)(23,24)(25,29)(26,27)(28,30)(31,32)(33,36)(34,35);(5,9)(6,13)(7,8)(10,19)(11,17)(12,15)(14,20)(16,18)(21,25)(22,29)(23,31)(24,32)(26,34)(27,35)(28,33)(30,36)

Polycyclic presentation:

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

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, 7673527);