#7108058

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1315 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7015254);