#321348

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

866 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)(5,6,7,11,8,10,12,9)(15,21)(18,26)(22,28)(23,25);(1,2,4,3)(13,14,17,20,16,19,24,27)(15,22,23,26,21,28,25,18);(6,9)(7,12)(10,11)(13,15,16,21)(14,18,19,26)(17,25,24,23)(20,28,27,22);(1,3,4,2)(6,10)(9,11)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 228544);