#356001

order	512
nilpotency class	3
isoclinism	isoclinic class of #17392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

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

2006 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)(15,18,22,25)(16,19)(17,20)(21,24,27,28)(23,26);(5,7,8,12)(6,9,10,11);(2,4)(7,12)(9,11)(13,15)(14,18,20,25)(16,21)(17,22)(19,24,26,28)(23,27);(6,10)(9,11)(13,16,17,23)(14,19,20,26)(15,21,22,27)(18,24,25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263197);