#182182

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1182 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)(7,13)(8,16)(9,12)(10,15)(11,17)(14,19)(18,20)(21,22)(23,26)(24,28)(25,27)(29,33)(30,32)(31,34)(35,36);(5,7)(6,10,12,18)(8,15)(9,14)(11,13,16,19)(17,20)(21,23,24,30,25,29,31,35)(22,26,28,32,27,33,34,36);(1,3,4,2)(22,27)(23,29)(28,34)(30,35);(7,14)(10,18)(13,19)(15,20)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89378);