#6388811

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂²×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#433 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1611 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,22,27,23,20,31,26)(19,24,29,38,30,32,43,37)(21,25,35,40,34,33,44,39)(28,36,42,47,41,45,48,46);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,10)(14,15)(17,19,22,29,23,30,31,43)(18,24,27,38,20,32,26,37)(21,28,35,42,34,41,44,48)(25,36,40,47,33,45,39,46);(17,20)(18,23)(19,24)(21,33)(22,26)(25,34)(27,31)(28,36)(29,38)(30,32)(35,39)(37,43)(40,44)(41,45)(42,47)(46,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,21)(18,25)(19,28)(20,33)(22,35)(23,34)(24,36)(26,39)(27,40)(29,42)(30,41)(31,44)(32,45)(37,46)(38,47)(43,48);(1,5,6,4)(2,8,9,14)(3,11,12,13)(7,16,10,15)(21,34)(25,33)(28,41)(35,44)(36,45)(39,40)(42,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6296007);