#7184010

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996799
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	#3041	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1575 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=h,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 7091206);