#7293216

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1103 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=gi,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,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, 7200412);