#7624484

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624170
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 ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1288 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)(19,30)(20,29)(21,27)(22,37)(23,26)(24,32)(25,33)(28,39)(31,41)(34,44)(35,45)(36,40)(38,46)(42,48)(43,47);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(17,19)(18,24)(20,34)(21,32)(22,33)(23,31)(25,40)(26,44)(27,30)(28,42)(29,41)(35,48)(36,47)(37,43)(38,39)(45,46);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,19)(18,25)(20,31)(21,32)(22,38)(23,34)(24,40)(26,42)(27,43)(28,44)(29,46)(30,37)(33,39)(35,47)(36,48)(41,45);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16);(2,10)(7,11)(8,9)(15,16)(17,20,21,23)(18,26,27,29)(19,31,32,34)(22,35,36,39)(24,41,30,44)(25,42,43,46)(28,40,45,37)(33,38,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319518, #72320155, #72320187
maximal normal subgroup of	#72319518, #72320155, #72320187

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531680);