#7656811

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

7196 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)(8,16)(11,18)(15,19)(17,20)(21,22,26,24)(23,29,31,27)(25,28,33,32)(30,35,36,34);(1,2)(3,4)(5,6)(7,14)(8,11)(9,12)(10,13)(15,20)(16,18)(17,19);(5,7,10,12)(6,9,13,14)(8,15,16,19)(11,17,18,20)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35);(5,8)(6,11)(7,15)(9,17)(10,16)(12,19)(13,18)(14,20);(5,6)(7,9)(8,11)(10,13)(12,14)(15,17)(16,18)(19,20)(21,24,26,22)(23,29,31,27)(25,28,33,32)(30,34,36,35);(5,9)(6,12)(7,13)(8,17)(10,14)(11,19)(15,18)(16,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7564007);