#7184009

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	#3508	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1573 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)(19,28,32,38)(20,25)(21,26)(22,27)(23,30,24,31)(29,40,44,47)(33,39,45,37)(34,41)(35,42)(36,43)(46,48);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19)(18,23)(20,33)(21,31)(22,32)(24,27)(25,40)(26,38)(28,43)(29,35)(30,36)(34,45)(37,42)(39,48)(41,47)(44,46);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,19)(18,24)(20,29)(21,30)(22,32)(23,27)(25,39)(26,38)(28,43)(31,36)(33,35)(34,44)(37,41)(40,48)(42,47)(45,46);(3,10)(6,9)(11,16)(14,15)(17,20,21,35,22,34,36,46)(18,25,26,42,27,41,43,48)(19,29,30,33,32,44,31,45)(23,37,28,40,24,39,38,47);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(20,34)(25,41)(29,44)(33,45)(35,46)(37,39)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=fgi,b^2=h,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, 7091205);