#6389890

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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2182 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)(7,10)(8,11)(9,12)(13,14,18,22)(15,23,27,37)(16,20,28,34)(17,30,24,21)(19,31,33,26)(25,36,39,44)(29,41,38,35)(32,42,43,40);(1,3,4,2)(5,7,10,6)(8,11,9,12);(13,15,24,31,18,27,17,26)(14,19,30,37,22,33,21,23)(16,25,38,42,28,39,29,40)(20,32,41,44,34,43,35,36);(5,8,10,9)(6,11,7,12)(13,16)(14,20)(15,25)(17,29)(18,28)(19,32)(21,35)(22,34)(23,36)(24,38)(26,40)(27,39)(30,41)(31,42)(33,43)(37,44);(5,9,10,8)(6,11,7,12)(16,28)(20,34)(25,39)(29,38)(32,43)(35,41)(36,44)(40,42)

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=gi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^d=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, 6297086);