#7624967

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
prime radical	#33813	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1174 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,31)(20,35)(21,27)(22,29)(23,42)(24,28)(25,33)(26,40)(30,43)(32,44)(34,37)(36,46)(38,48)(39,47)(41,45);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,25)(20,30)(21,26)(22,33)(23,34)(24,32)(27,44)(28,40)(29,31)(35,36)(37,45)(38,42)(39,46)(41,48)(43,47);(17,20,36,31,24,38,23,40,22,39,43,25,21,37,41,44)(18,26,45,48,27,19,30,35,29,32,42,34,28,33,46,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21,22,24)(18,27,29,28)(19,32,33,26)(20,37,39,38)(23,36,41,43)(25,40,31,44)(30,42,46,45)(34,47,48,35);(2,8)(6,9)(7,16)(14,15)(17,21,22,24)(18,28,29,27)(19,32,33,26)(20,38,39,37)(23,36,41,43)(25,44,31,40)(30,45,46,42)(34,47,48,35)

Polycyclic presentation:

2^9|b^2=fi,c^2=ghi,d^2=f,e^2=i,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,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, 7532163);