#180968

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1634 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=ci,e^a=eh,g^a=gh,c^b=ch

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238824
maximal normal subgroup of	#72238824

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88164);