#406278

order	512
nilpotency class	3
isoclinism	isoclinic class of #394329
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3055	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1913 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=f,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^b=fi,e^c=ei,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, 313474);