#7736968

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

7198 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7644164);