#7624495

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624235
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

1273 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18)(19,26)(20,31)(21,27)(22,35)(23,24)(25,33)(28,37)(29,45)(30,43)(32,47)(34,40)(36,41)(38,44)(39,48)(42,46);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(17,19,21,23)(18,24,27,26)(20,29,32,38)(22,30,34,37)(25,39,42,41)(28,40,43,35)(31,44,47,45)(33,36,46,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20)(18,25)(19,29)(21,32)(22,36)(23,38)(24,39)(26,41)(27,42)(28,44)(30,46)(31,35)(33,37)(34,48)(40,47)(43,45);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(17,19,21,23)(18,26,27,24)(20,29,32,38)(22,30,34,37)(25,41,42,39)(28,35,43,40)(31,45,47,44)(33,36,46,48);(2,10)(7,11)(8,9)(15,16)(18,27)(24,26)(25,42)(28,43)(31,47)(35,40)(39,41)(44,45)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287695, #72320169, #72320184
maximal normal subgroup of	#72287695, #72320169, #72320184

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531691);