#414994

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
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 ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1470 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:

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=i,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,d^c=di

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, 322190);