#267664

order	512
nilpotency class	3
isoclinism	isoclinic class of #267659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35150	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fh,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 174860);