#322578

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
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×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

827 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)(13,14,17,22,18,23,16,21)(15,19,24,27,26,28,25,20);(1,2,4,3)(5,6,9,10,8,12,7,11)(13,15,17,24,18,26,16,25)(14,19,22,27,23,28,21,20);(6,10)(7,9)(11,12)(13,15,18,26)(14,20,23,27)(16,24,17,25)(19,22,28,21);(1,3,4,2)(13,16,18,17)(14,21,23,22)(15,24,26,25)(19,27,28,20)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,c^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,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, 229774);