#324755

order	512
nilpotency class	3
isoclinism	isoclinic class of #279051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

591 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)(3,7)(4,5)(6,8)(17,18,22,27)(19,29,25,24)(20,31,32,30)(21,28,23,26);(2,5)(6,7)(9,10,12,14,13,15,11,16)(17,19)(18,24)(20,28)(21,31)(22,25)(23,30)(26,32)(27,29);(9,10)(11,14)(12,16)(13,15)(17,20,21,24,22,32,23,29)(18,25,26,30,27,19,28,31);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,14,15,16)(17,21,22,23)(18,26,27,28)(19,30,25,31)(20,29,32,24)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fh,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dhi,d^c=dh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 231951);