#412440

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	#60450	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1412 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)(5,6,11,14)(7,10,17,24)(8,18,22,12)(9,16,26,31)(13,28,29,25)(15,21,19,23)(20,33,35,30)(27,36,32,34);(5,7,10,14,11,17,24,6)(8,19,21,18,22,15,23,12)(9,16,25,29,26,31,28,13)(20,33,34,32,35,30,36,27);(2,4)(5,8)(6,12)(7,15)(9,20)(10,23)(11,22)(13,27)(14,18)(16,30)(17,19)(21,24)(25,36)(26,35)(28,34)(29,32)(31,33);(5,9,24,28,11,26,10,25)(6,13,17,31,14,29,7,16)(8,20,21,34,22,35,23,36)(12,27,19,33,18,32,15,30)

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=hi,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

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, 319636);