#352205

order	512
nilpotency class	3
isoclinism	isoclinic class of #352064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

796 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)(5,6,10,8)(7,13,15,20)(9,17,18,12)(11,19,16,14)(21,22,27,32)(23,33,34,29)(24,36,35,28)(25,31,26,30);(5,7,9,14,10,15,18,19)(6,11,12,13,8,16,17,20)(21,23,26,24,27,34,25,35)(22,28,31,29,32,36,30,33);(5,8,10,6)(7,11,15,16)(9,17,18,12)(13,19,20,14)(21,24,26,34,27,35,25,23)(22,29,31,36,32,33,30,28);(2,4)(6,8)(11,16)(12,17)(13,20)(23,34)(24,35)(28,36)(29,33)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=fhi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,c^b=ch,d^b=di,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259401);