#405760

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312956);