#6403896

order	512
nilpotency class	3
isoclinism	isoclinic class of #62265
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂×#548	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1852 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fgi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei

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, 6311092);