#6372476

order	512
nilpotency class	3
isoclinism	isoclinic class of #60460
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	#62814	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,18,15,19,13,28)(11,20,24,29,21,33,23,16)(12,17,27,32,25,31,26,38)(22,34,37,39,35,40,36,30);(9,11)(10,16,19,29)(12,22)(13,23)(14,24)(15,21)(17,30,31,39)(18,20,28,33)(25,35)(26,36)(27,37)(32,34,38,40);(2,5)(6,8)(11,21)(16,29)(20,33)(22,35)(23,24)(30,39)(34,40)(36,37);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,22)(13,26)(14,27)(15,25)(16,30)(18,32)(19,31)(20,34)(21,35)(23,36)(24,37)(28,38)(29,39)(33,40);(12,25)(17,31)(22,35)(26,27)(30,39)(32,38)(34,40)(36,37)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318196
maximal normal subgroup of	#72318196

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279672);