#296954

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

721 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204150);