#181526

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

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

1728 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)(7,14)(8,13)(9,18)(10,12)(11,17)(15,20)(16,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,7)(6,11,12,19)(8,16)(9,17)(10,15)(13,14,18,20)(21,23)(22,26,28,29)(24,30)(25,31)(27,33,34,35)(32,36);(6,12)(7,15)(13,18)(16,17);(5,8,10,9)(6,13,12,18)(7,16,15,17)(11,14,19,20)(21,24,25,32)(22,27,28,34)(23,30,31,36)(26,33,29,35)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=h,b^a=be,c^a=ch,e^a=eh,g^a=ghi,c^b=ch,e^b=ei

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, 88722);