#382493

order	512
nilpotency class	3
isoclinism	isoclinic class of #29610
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

2472 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)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,21)(20,25)(22,26)(23,28)(24,27);(1,2)(3,4)(6,9)(10,12)(13,15,17,23)(14,18,21,28)(16,24,25,22)(19,27,20,26);(2,4)(7,11)(10,12)(14,19,21,20)(18,26,28,27);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

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

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, 289689);