#411577

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2535	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2358 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|c^2=i,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,g^b=gi,g^c=gi

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, 318773);