#413498

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1110 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:

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

Polycyclic presentation:

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

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, 320694);