#204134

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1202 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)(7,13)(8,16)(9,11)(10,14)(12,17)(15,19)(18,20)(21,22,26,29,25,30,24,27)(23,31,34,35,32,28,33,36);(5,7,8,14,9,15,17,20)(6,10,12,13,11,18,16,19)(22,27,30,29)(28,35,31,36);(1,2,3,4)(6,11)(10,18)(12,16)(13,19)(21,23)(22,28)(24,33)(25,32)(26,34)(27,35)(29,36)(30,31);(23,32)(28,31)(33,34)(35,36)

Polycyclic presentation:

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

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, 111330);