#6461040

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	#28562	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1642 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)(17,18,23,28)(19,29,31,25)(20,22,27,24)(21,26,32,30);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,30)(21,24)(22,32)(23,31)(26,27)(28,29);(10,14)(11,15)(17,20,24,18,23,27,22,28)(19,30,21,25,31,26,32,29);(11,15)(13,16)(17,21)(18,26)(19,22)(20,25)(23,32)(24,31)(27,29)(28,30);(1,3,4,7)(2,6,5,8)(17,22,23,24)(18,27,28,20)(19,21,31,32)(25,30,29,26)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,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,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6368236);