#394380

order	512
nilpotency class	3
isoclinism	isoclinic class of #394378
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2644 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)(7,16)(8,12)(9,25)(10,15)(11,20)(13,23)(14,17)(18,31)(19,27)(21,33)(22,30)(24,29)(26,36)(28,34)(32,35);(5,7,17,34,10,21,9,20)(6,11,25,33,15,28,14,16)(8,19,23,35,24,18,22,36)(12,26,30,31,29,32,13,27);(2,4)(5,8)(6,12)(7,18)(9,23)(10,24)(11,26)(13,25)(14,30)(15,29)(16,31)(17,22)(19,21)(20,36)(27,33)(28,32)(34,35);(5,8)(6,13)(7,19)(9,22)(10,24)(11,27)(12,25)(14,29)(15,30)(16,32)(17,23)(18,21)(20,36)(26,33)(28,31)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316837
maximal normal subgroup of	#72316837

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301576);