#356010

order	512
nilpotency class	3
isoclinism	isoclinic class of #17405
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

1626 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)(8,10)(9,12)(13,14,18,16,17,21,19,26)(15,20,24,22,23,27,25,28);(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,22)(17,23)(18,24)(19,25)(21,27)(26,28);(2,4)(7,12)(9,11)(13,16,19,21,17,26,18,14)(15,22,25,27,23,28,24,20);(6,10)(9,11)(15,23)(20,27)(22,28)(24,25)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263206);