#397034

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
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	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1080 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)(7,11)(8,12)(9,10)(13,14)(15,25)(16,18)(17,28)(19,23)(20,24)(21,27)(22,26);(6,10)(11,12)(13,15,19,22,20,28,18,27)(14,21,16,17,24,26,23,25);(5,7,9,8)(6,11,10,12)(13,16,18,14,20,23,19,24)(15,26,27,17,28,21,22,25);(2,4)(5,8,9,7)(6,12,10,11)(13,17,18,21,20,25,19,26)(14,22,23,15,24,27,16,28)

Polycyclic presentation:

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

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, 304230);