#6372739

order	512
nilpotency class	3
isoclinism	isoclinic class of #60399
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#3025	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,d^2=hi,e^2=i,b^a=bf,c^a=cg,e^a=ei,g^b=gi,f^c=fi,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281693, #72283331, #72283410
maximal normal subgroup of	#72281693, #72283331, #72283410

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279935);