#319003

order	512
nilpotency class	3
isoclinism	isoclinic class of #262915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

538 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eghi,c^2=hi,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,f^c=fh,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, 226199);