#6461073

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1589 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)(17,18,21,23)(19,27,28,43)(20,25,33,37)(22,35,29,26)(24,38,36,31)(30,42,32,46)(34,48,44,41)(39,47,40,45);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(18,23)(19,28)(25,37)(26,35)(30,32)(31,38)(41,48)(45,47);(10,14)(11,15)(17,19,29,38,21,28,22,31)(18,24,35,43,23,36,26,27)(20,32,44,45,33,30,34,47)(25,40,48,42,37,39,41,46);(11,15)(13,16)(17,20)(18,25)(19,30)(21,33)(22,34)(23,37)(24,39)(26,41)(27,42)(28,32)(29,44)(31,45)(35,48)(36,40)(38,47)(43,46);(1,3,4,7)(2,6,5,8)(20,33)(25,37)(30,32)(34,44)(39,40)(41,48)(42,46)(45,47)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,d^c=di,e^d=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, 6368269);