#381479

order	512
nilpotency class	3
isoclinism	isoclinic class of #219508
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1355 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)(5,6,10,17,11,16,30,15)(7,18,22,32,23,36,21,12)(8,24,25,20,26,13,29,33)(9,14,27,34,28,35,31,19);(5,7)(6,12)(8,19)(9,20)(10,22)(11,23)(13,27)(14,25)(15,36)(16,32)(17,18)(21,30)(24,31)(26,34)(28,33)(29,35);(2,4)(5,8,11,26)(6,13,16,24)(7,19,23,34)(9,18,28,36)(10,29,30,25)(12,31,32,27)(14,22,35,21)(15,33,17,20);(5,9,10,27,11,28,30,31)(6,14,17,34,16,35,15,19)(7,20,22,13,23,33,21,24)(8,12,25,18,26,32,29,36)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,d^c=de,e^c=ei,h^c=hi,f^d=fi,g^d=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, 288675);