#281576

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

784 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)(13,14,18,20,17,21,16,22)(15,23,27,24,26,19,25,28);(2,4)(6,9)(7,11)(10,12)(13,15,16,25,17,26,18,27)(14,19,22,24,21,23,20,28);(5,6,8,10)(7,12,11,9)(13,14,17,21)(15,24,26,28)(16,20,18,22)(19,27,23,25);(2,4)(13,16,17,18)(14,20,21,22)(15,25,26,27)(19,28,23,24)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=hi,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, 188772);