#353834

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1770 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)(13,14,18,15,17,21,19,23)(16,20,25,22,24,27,26,28);(2,4)(13,15,19,21,17,23,18,14)(16,22,26,27,24,28,25,20);(6,9)(10,12)(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28);(5,7,8,11)(6,10,9,12)(16,24)(20,27)(22,28)(25,26)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261030);