#183136

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1142 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285505
maximal normal subgroup of	#72285505

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90332);