#352188

order	512
nilpotency class	3
isoclinism	isoclinic class of #352174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

805 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)(7,13)(8,16)(9,11)(10,15)(12,17)(14,19)(18,20)(23,29)(24,30)(26,32)(28,33)(31,35)(34,36);(5,7,8,15,9,14,17,20)(6,10,12,13,11,18,16,19)(21,22,25,27)(23,26,31,34)(24,28,30,33)(29,32,35,36);(6,11)(7,14)(12,16)(15,20)(21,23,30,35,25,31,24,29)(22,26,33,36,27,34,28,32);(2,4)(22,27)(26,34)(28,33)(32,36)

Polycyclic presentation:

2^9|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

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, 259384);