#7624390

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
prime radical	#33669	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1226 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)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,28)(20,25)(21,24)(22,35)(23,29)(26,34)(27,33)(30,36)(31,38)(32,40)(37,46)(39,44)(41,48)(42,47)(43,45);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,20,29)(18,23,25,28)(21,30,32,44)(22,31,33,45)(24,39,40,36)(26,37,41,42)(27,38,35,43)(34,47,48,46);(18,24)(19,29)(21,34)(22,33)(23,36)(25,40)(26,27)(28,39)(30,46)(32,48)(35,41)(37,43)(38,42)(44,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(19,29)(23,28)(30,44)(31,45)(36,39)(37,42)(38,43)(46,47);(2,8)(6,9)(7,16)(14,15)(18,25)(23,28)(24,40)(26,41)(27,35)(36,39)(37,42)(38,43)

Polycyclic presentation:

2^9|b^2=f,d^2=fi,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	#72318554, #72320129, #72320233
maximal normal subgroup of	#72318554, #72320129, #72320233

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531586);