#369417

order	512
nilpotency class	3
isoclinism	isoclinic class of #196038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=efg,d^2=ei,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276613);