#382279

order	512
nilpotency class	3
isoclinism	isoclinic class of #29742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=e,d^2=efh,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=de,g^c=gi

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, 289475);