#6461018

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1594 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)(19,24)(20,32)(21,26)(22,37)(23,27)(25,36)(28,43)(29,39)(30,40)(31,46)(33,47)(34,42)(35,48)(38,45)(41,44);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,33)(21,29)(22,31)(23,30)(25,41)(26,39)(27,40)(28,35)(32,47)(34,45)(36,44)(37,46)(38,42)(43,48);(10,14)(11,15)(17,20,21,34,23,35,22,36)(18,25,37,48,27,42,26,32)(19,28,29,44,30,33,31,45)(24,38,46,47,40,41,39,43);(11,15)(13,16)(17,21,23,22)(18,26,27,37)(19,29,30,31)(20,34,35,36)(24,39,40,46)(25,32,42,48)(28,44,33,45)(38,43,41,47);(1,3,4,7)(2,6,5,8)(19,30)(20,35)(24,40)(25,42)(29,31)(32,48)(34,36)(39,46)

Polycyclic presentation:

2^9|c^2=gh,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6368214);