#7512119

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1544 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)(19,25)(20,26)(21,27)(22,37)(23,28)(24,29)(30,39)(31,40)(32,42)(33,43)(34,44)(35,46)(36,45)(38,41)(47,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,24,33)(18,25,29,43)(20,34,21,31)(22,32,23,38)(26,44,27,40)(28,41,37,42)(30,36,47,35)(39,45,48,46);(10,14)(13,15)(17,20,24,21)(18,26,29,27)(19,30,33,47)(22,35,23,36)(25,39,43,48)(28,45,37,46)(31,38,34,32)(40,41,44,42);(2,5)(6,8)(11,16)(13,15)(17,21,24,20)(18,27,29,26)(19,31,33,34)(22,35,23,36)(25,40,43,44)(28,45,37,46)(30,38,47,32)(39,41,48,42);(1,3,4,7)(2,6,5,8)(17,22,24,23)(18,28,29,37)(19,32,33,38)(20,35,21,36)(25,41,43,42)(26,45,27,46)(30,31,47,34)(39,44,48,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,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, 7419315);