#384442

order	512
nilpotency class	3
isoclinism	isoclinic class of #232138
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

959 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^c=hi,h^d=hi

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, 291638);