#213560

order	512
nilpotency class	3
isoclinism	isoclinic class of #213555
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2635	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22)(25,26,29,33,30,32,28,36)(27,31,35,40,34,39,37,38);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,27)(26,31)(28,37)(29,35)(30,34)(32,39)(33,40)(36,38);(3,8)(5,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23)(26,32)(27,34)(28,29)(38,40);(2,6)(5,7)(11,18)(14,22)(17,24)(21,23)(25,28,30,29)(26,33,32,36)(27,35,34,37)(31,38,39,40)

Polycyclic presentation:

2^9|a^2=g,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cei,d^b=dh,d^c=di,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120756);