#381904

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=efh,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,d^c=de,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289100);