#360323

order	512
nilpotency class	3
isoclinism	isoclinic class of #25196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1837 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:

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

Polycyclic presentation:

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

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, 267519);