#288748

order	512
nilpotency class	3
isoclinism	isoclinic class of #288715
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	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

747 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=ef,c^2=i,d^2=ei,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^c=fi,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, 195944);