#7613018

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	#45243	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1553 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,28,35)(20,24,33,40)(21,25,32,39)(27,34,31,38)(29,36,42,46)(30,37,44,48)(41,45,43,47);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(20,30)(21,32)(24,37)(25,39)(27,41)(29,42)(31,43)(33,44)(34,45)(36,46)(38,47)(40,48);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,19)(18,23)(20,31)(21,29)(22,28)(24,38)(25,36)(26,35)(27,33)(30,43)(32,42)(34,40)(37,47)(39,46)(41,44)(45,48);(10,14)(13,16)(17,20,32,44,22,33,21,30)(18,24,39,48,26,40,25,37)(19,27,42,43,28,31,29,41)(23,34,46,47,35,38,36,45);(1,4,5,3)(2,7,8,6)(19,28)(20,33)(23,35)(24,40)(29,42)(30,44)(36,46)(37,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=hi,e^2=f,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, 7520214);