#360825

order	512
nilpotency class	3
isoclinism	isoclinic class of #27987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
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

1180 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268021);