#7624392

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624128
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 ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1216 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319166, #72320055, #72320309
maximal normal subgroup of	#72319166, #72320055, #72320309

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531588);