#205316

order	512
nilpotency class	3
isoclinism	isoclinic class of #19328
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1026 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)(5,6,10,13,9,12,8,14)(7,15,16,11,18,20,17,19)(21,22,26,30,27,29,25,23)(24,32,34,36,35,31,33,28);(1,2,3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20)(21,23,26,29,27,30,25,22)(24,31,34,36,35,32,33,28);(6,12)(11,19)(13,14)(15,20)(21,24,25,33,27,35,26,34)(22,28,30,32,29,36,23,31);(5,8,9,10)(6,13,12,14)(7,16,18,17)(11,15,19,20)(22,29)(23,30)(28,36)(31,32)

Polycyclic presentation:

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

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, 112512);