#298122

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1218 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:

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

Polycyclic presentation:

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

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, 205318);