#7608505

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23871	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1665 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,29,32,24)(20,34,31,25)(21,38,40,26)(23,28,41,47)(30,42,35,44)(33,46,36,43)(37,48,39,45);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,32)(18,24,27,29)(20,35,31,30)(21,39,40,37)(23,36,41,33)(25,44,34,42)(26,45,38,48)(28,46,47,43);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20)(18,25)(19,30)(21,33)(22,31)(23,37)(24,42)(26,46)(27,34)(28,45)(29,44)(32,35)(36,40)(38,43)(39,41)(47,48);(10,14)(13,16)(17,21,23,35,22,40,41,30)(18,26,28,42,27,38,47,44)(19,31,33,39,32,20,36,37)(24,25,43,45,29,34,46,48);(2,5)(6,8)(18,27)(20,31)(21,40)(24,29)(28,47)(30,35)(37,39)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=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, 7515701);