#267759

order	512
nilpotency class	3
isoclinism	isoclinic class of #267755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 174955);