#387408

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
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

1930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,4)(5,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,26)(20,24,30,23)(21,25)(22,29)(27,31,28,32);(1,3)(2,6)(4,7)(5,8)(10,14)(11,12)(17,19)(18,22)(20,27)(21,29)(23,31)(24,32)(25,26)(28,30);(1,4,5,6)(2,3,7,8)(9,11,13,12)(10,15,14,16)(17,20)(18,23)(19,27)(21,30)(22,31)(24,25)(26,32)(28,29);(2,7)(3,8)(9,12,13,11)(10,15,14,16)(17,20)(18,24)(19,28)(21,30)(22,31)(23,25)(26,32)(27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294604);