#360826

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ef,d^2=ei,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, 268022);