#393984

order	512
nilpotency class	3
isoclinism	isoclinic class of #393982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62852	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4040 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:

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

Polycyclic presentation:

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

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, 301180);