#394023

order	512
nilpotency class	3
isoclinism	isoclinic class of #394022
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3960 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)(6,12)(9,10)(13,17)(14,29)(15,16)(19,20)(22,24)(25,30)(26,35)(27,28)(31,36)(32,34);(5,6)(7,17)(8,14)(9,15)(10,12)(11,16)(13,20)(18,31)(19,28)(21,27)(22,25)(23,30)(24,29)(26,32)(33,35)(34,36);(2,4)(5,7,11,21)(6,13,16,28)(8,18,23,33)(9,20,10,19)(12,17,15,27)(14,26,30,36)(22,34,24,32)(25,31,29,35);(5,8)(6,14)(7,18)(9,24)(10,22)(11,23)(12,25)(13,26)(15,29)(16,30)(17,31)(19,34)(20,32)(21,33)(27,35)(28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239341, #72286934, #72287183, …
maximal normal subgroup of	#72239341, #72286934, #72287183, …
maximal quotient of	#72280705, #72316704

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301219);