#205444

order	512
nilpotency class	3
isoclinism	isoclinic class of #19552
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

996 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)(7,15)(8,10)(9,13)(11,14)(12,18)(16,19)(17,20)(21,22,25,30,26,31,24,29)(23,28,32,35,33,36,34,27);(1,2,3,4)(5,7,10,18,11,19,9,17)(6,12,13,15,14,20,8,16)(21,23)(22,27)(24,32)(25,34)(26,33)(28,29)(30,36)(31,35);(5,8,9,14,11,13,10,6)(7,16,17,20,19,15,18,12)(21,23)(22,28)(24,34)(25,32)(26,33)(27,29)(30,35)(31,36);(5,9,11,10)(6,13,14,8)(7,17,19,18)(12,15,20,16)(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=hi,e^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,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, 112640);