#204758

order	512
nilpotency class	3
isoclinism	isoclinic class of #7654
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1170 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=hi,e^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111954);