#7642110

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

6476 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi,d^c=dhi

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, 7549306);