#7624359

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×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1312 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)(19,23)(20,24)(21,34)(22,27)(25,31)(26,32)(28,36)(29,35)(30,44)(33,40)(37,43)(38,42)(39,45)(41,46)(47,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,22,28)(18,23,27,36)(20,29,33,41)(21,30,32,42)(24,35,40,46)(25,37,39,47)(26,38,34,44)(31,43,45,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,24)(20,31)(21,32)(23,35)(25,26)(27,40)(29,43)(30,42)(33,45)(34,39)(36,46)(37,38)(41,48)(44,47);(3,11)(6,9)(10,16)(13,15)(19,28)(23,36)(29,41)(30,42)(35,46)(37,47)(38,44)(43,48);(2,8)(6,9)(7,14)(13,15)

Polycyclic presentation:

2^9|b^2=i,c^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	#72287767, #72320020, #72320482
maximal normal subgroup of	#72287767, #72320020, #72320482

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531555);