#399097

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1640 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)(11,16)(12,14)(13,15)(17,18,22,27)(19,28,20,31)(21,25,32,26)(23,29,24,30);(1,2)(3,4)(9,11,13,12)(10,14,15,16)(17,19,22,20)(18,23,27,24)(21,29,32,30)(25,28,26,31);(2,4)(9,12,13,11)(10,14,15,16)(17,20,22,19)(18,24,27,23)(21,30,32,29)(25,31,26,28);(6,8)(17,21,22,32)(18,25,27,26)(19,29,20,30)(23,28,24,31)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 306293);