#6380912

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,27)(17,31)(18,21)(19,25)(20,30)(22,34)(24,37)(26,39)(28,35)(29,38)(32,36)(33,40)(41,42)(43,44);(13,15,18,26)(14,19,21,36)(16,28,30,43)(17,32,34,25)(20,37,27,41)(22,39,31,23)(24,33,42,29)(35,40,44,38);(6,9)(10,12);(13,16,18,30)(14,20,21,27)(15,24,26,42)(17,33,34,29)(19,35,36,44)(22,40,31,38)(23,41,39,37)(25,43,32,28);(5,7,8,11)(6,10,9,12)(14,21)(19,36)(20,27)(22,31)(23,39)(35,44)(37,41)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6288108);