#6460852

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1649 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,24)(19,21,26,23)(20,25,28,32)(27,31,29,30);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(18,24)(19,26)(20,28)(30,31);(10,14)(11,15)(17,19,23,18,22,26,21,24)(20,29,30,32,28,27,31,25);(11,15)(13,16)(17,20)(18,25)(19,27)(21,31)(22,28)(23,30)(24,32)(26,29);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,26,24,19)(20,30,28,31)(25,27,32,29)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6368048);