#298229

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
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
prime radical	C₂×#550	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1208 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)(7,11)(8,9)(13,14)(15,21)(16,19)(17,26)(18,23)(20,25)(22,27)(24,28);(1,2,4,3)(6,9)(7,12)(10,11)(13,15,16,23)(14,18,19,21)(17,27,25,24)(20,22,26,28);(5,7,6,10,8,11,9,12)(13,14)(15,22,23,28)(16,19)(17,20)(18,27,21,24)(25,26);(1,3,4,2)(5,6,8,9)(7,10,11,12)(14,19)(18,21)(20,26)(22,28)

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=ehi,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=dhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205425);