#396419

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1269 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)(5,6,9,11)(7,12,8,10)(13,14,16,20)(15,23,24,18)(17,21,27,22)(19,25,28,26);(5,7,9,8)(6,10,11,12)(13,14,16,20)(15,19,24,28)(17,22,27,21)(18,26,23,25);(2,4)(13,15)(14,18)(16,24)(17,26)(19,22)(20,23)(21,28)(25,27);(5,8,9,7)(6,10,11,12)(13,15)(14,19)(16,24)(17,25)(18,22)(20,28)(21,23)(26,27)

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,g^c=gi,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, 303615);