#205466

order	512
nilpotency class	3
isoclinism	isoclinic class of #19456
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

1036 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)(7,12)(8,17)(9,19)(10,14)(11,16)(13,18)(15,20)(21,22,25,30,26,31,24,29)(23,28,32,35,33,36,34,27);(1,2,3,4)(5,7,9,16,10,15,18,8)(6,11,13,12,14,17,19,20)(21,23)(22,27)(24,32)(25,34)(26,33)(28,29)(30,36)(31,35);(5,8,18,15,10,16,9,7)(6,12,19,11,14,20,13,17)(21,23)(22,28)(24,34)(25,32)(26,33)(27,29)(30,35)(31,36);(7,15)(8,16)(11,17)(12,20)(21,24,26,25)(22,29,31,30)(23,32,33,34)(27,28,35,36)

Polycyclic presentation:

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

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, 112662);