#7655100

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

6812 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)(5,6,11,9)(7,14,12,8)(10,13,18,17)(15,20,19,16)(21,22,26,29)(23,27,30,34)(24,25,28,33)(31,32,35,36);(1,2,3,4)(23,30)(27,34)(31,35)(32,36);(5,7)(6,8)(9,14)(10,15)(11,12)(13,16)(17,20)(18,19)(21,23,26,30)(22,27,29,34)(24,31,28,35)(25,32,33,36);(5,8,11,14)(6,12,9,7)(10,16,18,20)(13,19,17,15);(5,9,11,6)(7,14,12,8)(10,13,18,17)(15,16,19,20)(21,24)(22,25)(23,31)(26,28)(27,32)(29,33)(30,35)(34,36);(5,10,11,18)(6,13,9,17)(7,15,12,19)(8,16,14,20)(21,25)(22,28)(23,32)(24,29)(26,33)(27,35)(30,36)(31,34)

Polycyclic presentation:

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

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, 7562296);