#398598

order	512
nilpotency class	3
isoclinism	isoclinic class of #394177
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1556 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305794);