#433409

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
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

1078 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)(7,9)(8,10)(11,12)(13,14,19,23,20,17,18,22)(15,16,26,28,21,24,25,27);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15,18,25,20,21,19,26)(14,16,22,27,17,24,23,28);(1,2,4,3)(6,10)(9,12)(13,16)(14,21)(15,17)(18,28)(19,27)(20,24)(22,25)(23,26);(1,3,4,2)(13,17,20,14)(15,24,21,16)(18,23,19,22)(25,28,26,27)

Polycyclic presentation:

2^9|a^2=eg,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,d^b=de,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, 340605);