#204135

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂²×C₂²⋊C₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1240 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215872, #72216040, #72216055
maximal normal subgroup of	#72215872, #72216040, #72216055

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111331);