#6816565

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3355	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=fi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282995, #72283215, #72283271
maximal normal subgroup of	#72282995, #72283215, #72283271

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723761);