#353777

order	512
nilpotency class	2
isoclinism	isoclinic class of #180864
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,h^a=hi,g^b=gi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226739, #72226742
maximal normal subgroup of	#72226739, #72226742

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260973);