#7624685

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
prime radical	#61431	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1246 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:

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

Polycyclic presentation:

2^9|a^2=fi,e^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	#72318554, #72320259, #72320285
maximal normal subgroup of	#72318554, #72320259, #72320285

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531881);