#7620954

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1292 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)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,23,28)(19,30,32,24)(20,34,38,25)(21,40,33,26)(22,29,41,27)(31,42,35,43)(36,44,47,48)(37,45,39,46);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,32)(18,24,28,30)(20,35,38,31)(21,41,33,22)(25,43,34,42)(26,27,40,29)(36,39,47,37)(44,45,48,46);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,25)(19,31)(21,36)(22,39)(23,38)(24,42)(26,44)(27,46)(28,34)(29,45)(30,43)(32,35)(33,47)(37,41)(40,48);(10,13)(14,16)(17,21,22,19,23,33,41,32)(18,26,29,24,28,40,27,30)(20,36,39,31,38,47,37,35)(25,44,45,42,34,48,46,43);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,22,23,41)(18,27,28,29)(19,21,32,33)(20,37,38,39)(24,40,30,26)(25,45,34,46)(31,47,35,36)(42,44,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=ghi,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7528150);