#7184013

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	#2018	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1577 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)(18,23,27,26)(19,28,31,30)(24,37)(25,32,39,40)(29,42,44,45)(36,41)(38,47)(43,48);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,18)(19,24)(20,32)(21,26)(22,27)(23,35)(25,34)(28,36)(29,43)(30,41)(31,37)(33,40)(38,45)(39,46)(42,47)(44,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,19,22,31)(18,24,27,37)(20,29,33,44)(21,30,35,28)(23,36,26,41)(25,38,39,47)(32,43,40,48)(34,45,46,42);(3,10)(6,9)(11,16)(14,15)(17,20,21,34,22,33,35,46)(18,25,23,32,27,39,26,40)(19,29,30,45,31,44,28,42)(24,38,36,43,37,47,41,48);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(20,33)(25,39)(29,44)(32,40)(34,46)(38,47)(42,45)(43,48)

Polycyclic presentation:

2^9|a^2=fgi,b^2=h,c^2=i,d^2=f,e^2=h,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, 7091209);