#6811991

order	512
nilpotency class	3
isoclinism	isoclinic class of #6807031
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3060	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1549 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=h,d^2=f,e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6719187);