#472054

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=efi,b^2=egh,c^2=i,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,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, 379250);