#416139

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#459	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1832 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,d^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, 323335);