#210966

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	#1999	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=chi,d^c=dhi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320225
maximal normal subgroup of	#72320225
maximal quotient of	#72320718, #72320764

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 118162);