#396448

order	512
nilpotency class	3
isoclinism	isoclinic class of #394048
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1422 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:

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

Polycyclic presentation:

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

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, 303644);