#204187

order	512
nilpotency class	3
isoclinism	isoclinic class of #195954
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1208 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)(7,10,14,16)(8,12)(9,11)(13,17,19,20)(15,18)(21,22,24,28,25,27,31,34)(23,26,29,33,30,32,35,36);(5,7)(6,10,11,16)(8,13)(9,14)(12,17,18,20)(15,19)(21,23,24,29,25,30,31,35)(22,26,28,33,27,32,34,36);(1,2,3,4)(6,11)(10,16)(12,18)(17,20);(5,8,9,15)(6,12,11,18)(7,13,14,19)(10,17,16,20)(22,27)(26,32)(28,34)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111383);