#398437

order	512
nilpotency class	3
isoclinism	isoclinic class of #398436
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60453	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1589 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=e,c^2=fi,d^2=e,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,f^b=fi,g^c=gi

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, 305633);