#408613

order	512
nilpotency class	3
isoclinism	isoclinic class of #393958
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1462 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315809);