#7020058

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45093	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1944 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6927254);