#7624324

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624184
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 ◁ #525 ◁ #2929 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1270 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)(17,18,22,27)(19,31,20,25)(21,30,24,28)(23,38,37,41)(26,33,44,35)(29,36,46,40)(32,42,34,47)(39,43,48,45);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,22,20)(18,25,27,31)(21,34,24,32)(23,33,37,35)(26,41,44,38)(28,47,30,42)(29,43,46,45)(36,39,40,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,22,19)(18,26,27,44)(21,34,24,32)(23,39,37,48)(25,41,31,38)(28,45,30,43)(29,42,46,47)(33,40,35,36);(3,11)(6,9)(10,16)(13,15)(18,27)(25,31)(26,44)(28,30)(29,46)(38,41)(42,47)(43,45);(2,8)(6,9)(7,14)(13,15)(17,21,22,24)(18,28,27,30)(19,32,20,34)(23,36,37,40)(25,42,31,47)(26,45,44,43)(29,41,46,38)(33,48,35,39)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,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

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319518, #72320482, #72320497
maximal normal subgroup of	#72319518, #72320482, #72320497

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531520);