#7625178

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624888
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 ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgh,d^2=i,e^2=fi,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,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532374);