#395089

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1886 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,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, 302285);