#7623514

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623305
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1321 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)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,34)(22,25)(26,31)(27,32)(28,35)(29,44)(30,36)(33,39)(37,41)(38,42)(40,45)(43,46)(47,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,22,30)(18,23,25,36)(20,28,33,43)(21,29,32,42)(24,35,39,46)(26,37,40,47)(27,38,34,44)(31,41,45,48);(3,10)(6,9)(11,16)(13,15)(18,24)(20,31)(21,32)(23,35)(25,39)(26,27)(28,41)(29,42)(33,45)(34,40)(36,46)(37,38)(43,48)(44,47);(2,7)(6,9)(8,14)(13,15);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(18,25)(23,36)(24,39)(26,40)(27,34)(35,46)(37,47)(38,44)

Polycyclic presentation:

2^9|b^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gh,h^a=hi,c^b=cf,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287478, #72287487, #72287492, …
maximal normal subgroup of	#72287478, #72287487, #72287492, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530710);