#7623516

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

1316 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)(19,28)(20,25)(21,33)(22,37)(23,26)(24,30)(27,35)(29,44)(31,45)(32,42)(34,39)(36,47)(38,43)(40,46)(41,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15);(3,10)(6,9)(11,16)(13,15)(17,19)(18,24)(20,29)(21,34)(22,38)(23,31)(25,39)(26,40)(27,43)(28,33)(30,35)(32,46)(36,48)(37,41)(42,44)(45,47);(2,7)(6,9)(8,14)(13,15)(17,20,23,22)(18,25,26,37)(19,29,31,38)(21,32,36,35)(24,39,40,41)(27,33,42,47)(28,44,45,43)(30,34,46,48);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(18,26)(24,40)(25,37)(27,42)(28,45)(33,47)(39,41)(43,44)

Polycyclic presentation:

2^9|d^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	#72287781, #72287783, #72287785, …
maximal normal subgroup of	#72287781, #72287783, #72287785, …
maximal quotient of	#72343806

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530712);