#288293

order	512
nilpotency class	3
isoclinism	isoclinic class of #264283
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

730 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 195489);