#368425

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1308 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=g,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275621);