#7023430

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	#45136	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1620 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)(11,13)(12,14)(15,16)(17,18,22,26)(19,27,28,39)(20,24,35,42)(21,36,29,25)(23,31,37,32)(30,43,44,48)(33,41,34,40)(38,46,47,45);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(19,28)(23,37)(27,39)(30,44)(31,32)(38,47)(43,48)(45,46);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,19,29,31,22,28,21,32)(18,23,36,39,26,37,25,27)(20,30,33,46,35,44,34,45)(24,38,40,48,42,47,41,43);(1,4,5,3)(2,7,6,8)(11,16)(13,15)(17,20,22,35)(18,24,26,42)(19,30,28,44)(21,34,29,33)(23,38,37,47)(25,41,36,40)(27,43,39,48)(31,46,32,45);(10,14)(13,15)(17,21,22,29)(18,25,26,36)(19,31,28,32)(20,33,35,34)(23,39,37,27)(24,40,42,41)(30,45,44,46)(38,43,47,48)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=fi,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=dhi,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, 6930626);