#267159

order	512
nilpotency class	3
isoclinism	isoclinic class of #267131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1032 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)(5,6,9,11,8,10,7,12)(13,14,18,20,17,21,16,22)(15,23,26,28,25,19,24,27);(2,4)(6,10)(11,12)(13,15,17,25)(14,19,21,23)(16,24,18,26)(20,27,22,28);(5,6)(7,11)(8,10)(9,12)(14,20)(15,24)(16,18)(21,22)(25,26)(27,28);(2,4)(5,7,8,9)(6,11,10,12)(13,16,17,18)(14,20,21,22)(15,24,25,26)(19,27,23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 174355);