#444293

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

1312 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gi,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 351489);