#213624

order	512
nilpotency class	3
isoclinism	isoclinic class of #163012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2847	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1127 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)(9,10,13,18,14,17,12,16)(11,19,21,24,20,15,22,23)(25,26)(27,31)(28,36,37,40)(29,34)(30,33)(32,35,39,38);(1,3)(2,5)(4,8)(6,7)(11,20)(15,19)(21,22)(23,24)(25,27,30,29)(26,31,33,34)(28,35,37,38)(32,36,39,40);(3,8)(5,7)(9,11)(10,15)(12,21)(13,22)(14,20)(16,23)(17,19)(18,24)(25,28)(26,32)(27,35)(29,38)(30,37)(31,36)(33,39)(34,40);(2,6)(5,7)(9,12,14,13)(10,16,17,18)(11,21,20,22)(15,23,19,24)(26,33)(31,34)(32,39)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ceh,f^c=fi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120820);