#7624327

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
prime radical	#62324	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1242 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)(19,27)(20,24)(21,33)(22,36)(23,29)(25,35)(26,32)(28,45)(30,47)(31,44)(34,42)(37,41)(38,43)(39,48)(40,46);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18,20,24)(19,23,28,38)(21,25,31,40)(22,26,32,36)(27,43,45,29)(30,39,37,42)(33,46,44,35)(34,41,48,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19,20,28)(18,23,24,38)(21,34,31,48)(22,37,32,30)(25,41,40,47)(26,42,36,39)(27,44,45,33)(29,46,43,35);(3,11)(6,9)(10,16)(13,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,38)(30,39)(31,40)(32,36)(33,35)(34,41)(37,42)(43,45)(44,46)(47,48);(2,8)(6,9)(7,14)(13,15)(18,24)(23,38)(25,40)(26,36)(27,45)(33,44)(39,42)(41,47)

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=df,e^b=ei,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531523);