#6458245

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437415
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6365441);