#392686

order	512
nilpotency class	3
isoclinism	isoclinic class of #196092
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3733 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299882);