#6489236

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470374
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1552 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,18,22,25)(19,27,29,43)(20,24,32,41)(21,34,35,26)(23,36,38,30)(28,42,31,46)(33,48,44,40)(37,47,39,45);(11,12)(14,15)(17,19,21,30,22,29,35,36)(18,23,26,27,25,38,34,43)(20,31,33,47,32,28,44,45)(24,39,40,46,41,37,48,42);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,16,15)(17,20)(18,24)(19,28)(21,33)(22,32)(23,37)(25,41)(26,40)(27,42)(29,31)(30,45)(34,48)(35,44)(36,47)(38,39)(43,46);(9,12,13,11)(10,15,16,14)(19,29)(23,38)(27,43)(28,31)(30,36)(37,39)(42,46)(45,47);(1,4,5,3)(2,6,7,8)(10,16)(14,15)(18,25)(20,32)(23,38)(26,34)(27,43)(28,31)(33,44)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=ghi,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^c=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, 6396432);