#455396

order	512
nilpotency class	3
isoclinism	isoclinic class of #454982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1187 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362592);