#397865

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1092 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=e,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^c=fi,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, 305061);