#7184012

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	#535	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1554 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)(19,31,36,43)(20,37,33,45)(21,27)(22,28)(23,29)(24,30)(25,34,44,35)(26,38,46,39)(32,48)(40,42)(41,47);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19)(18,25)(20,32)(21,38)(22,34)(23,35)(24,36)(26,42)(27,45)(28,31)(29,43)(30,44)(33,40)(37,47)(39,41)(46,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20)(18,26)(19,32)(21,34)(22,38)(23,39)(24,33)(25,42)(27,43)(28,37)(29,45)(30,46)(31,47)(35,41)(36,40)(44,48);(3,10)(6,9)(11,16)(14,15)(17,21,22,32,24,41,23,40)(18,27,28,48,30,47,29,42)(19,33,35,39,36,20,34,38)(25,26,43,37,44,46,31,45);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(17,22,24,23)(18,28,30,29)(19,34,36,35)(20,38,33,39)(21,40,41,32)(25,31,44,43)(26,37,46,45)(27,42,47,48)

Polycyclic presentation:

2^9|a^2=fgi,b^2=h,d^2=fi,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, 7091208);