#7512252

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1540 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,h^c=hi,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, 7419448);