#7184016

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	#26648	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=h,c^2=i,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, 7091212);