#387620

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

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

1358 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=efi,d^2=fg,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294816);