#214545

order	512
nilpotency class	3
isoclinism	isoclinic class of #162662
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	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1120 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)(28,35)(29,38)(30,33)(32,34)(36,40)(37,39);(1,3)(2,5)(4,8)(6,7)(11,20)(15,19)(21,22)(23,24)(25,27,30,29)(26,31,33,38)(28,36,37,34)(32,35,40,39);(3,8)(5,7)(9,11,14,20)(10,15,17,19)(12,21,13,22)(16,23,18,24)(25,28,30,37)(26,32,33,40)(27,34,29,36)(31,39,38,35);(2,6)(5,7)(9,12,14,13)(10,16,17,18)(11,21,20,22)(15,23,19,24)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121741);