#439387

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1210 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)(6,9)(10,11)(13,14,17,19)(15,18,16,20)(22,23)(26,27);(5,6,8,9)(7,11,12,10)(13,15,17,16)(14,18,19,20)(21,22,24,27,25,23,28,26);(13,16,17,15)(14,18,19,20)(21,23)(22,25)(24,27)(26,28);(2,4)(5,7,8,12)(6,10,9,11)(21,24,25,28)(22,26,23,27)

Polycyclic presentation:

2^9|a^2=fi,b^2=efg,c^2=fi,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 346583);