#367094

order	512
nilpotency class	3
isoclinism	isoclinic class of #24808
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1414 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efgh,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,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, 274290);