#7624479

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=f,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, #72320187, #72320286
maximal normal subgroup of	#72319518, #72320187, #72320286

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531675);