#359642

order	512
nilpotency class	3
isoclinism	isoclinic class of #188252
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3486	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3800 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:

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

Polycyclic presentation:

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

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, 266838);