#7608507

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

1672 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)(19,24,30,41)(20,32,35,25)(21,37,39,26)(23,28,38,46)(29,43,33,40)(31,42,34,44)(36,47,48,45);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,30)(18,24,27,41)(20,33,35,29)(21,36,39,48)(23,34,38,31)(25,43,32,40)(26,45,37,47)(28,44,46,42);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20,22,35)(18,25,27,32)(19,29,30,33)(21,31,39,34)(23,36,38,48)(24,40,41,43)(26,42,37,44)(28,45,46,47);(10,14)(13,16)(17,21,38,33,22,39,23,29)(18,26,46,43,27,37,28,40)(19,20,34,48,30,35,31,36)(24,25,44,47,41,32,42,45);(2,5)(6,8)(20,35)(21,39)(25,32)(26,37)(29,33)(36,48)(40,43)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=hi,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, 7515703);