#380241

order	512
nilpotency class	3
isoclinism	isoclinic class of #29610
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1946 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)(7,10)(8,9)(11,12)(13,14,18,21)(15,19,24,27)(16,25,20,17)(22,23,28,26);(6,9)(10,12)(13,15,18,24)(14,19,21,27)(16,22,20,28)(17,26,25,23);(5,7)(6,10)(8,11)(9,12)(13,16,17,14,18,20,25,21)(15,22,23,27,24,28,26,19);(7,11)(10,12)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,f^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=de,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, 287437);