#281572

order	512
nilpotency class	3
isoclinism	isoclinic class of #281531
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

772 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)(13,14,17,21,19,22,18,15)(16,24,27,20,26,23,25,28);(2,4)(6,9)(7,11)(10,12)(13,15,18,22,19,21,17,14)(16,20,25,24,26,28,27,23);(5,6,8,10)(7,12,11,9)(13,16)(14,20)(15,23)(17,25)(18,27)(19,26)(21,24)(22,28);(2,4)(13,17,19,18)(14,21,22,15)(16,25,26,27)(20,24,28,23)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=i,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 188768);