#477348

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

921 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14,17,22,18,23,34,38)(15,24,27,37,28,42,32,19)(16,20,30,39,31,40,33,21)(25,43,26,36,29,41,44,35);(6,9)(10,11)(13,15,18,28)(14,19,23,37)(16,29,31,25)(17,32,34,27)(20,36,40,35)(21,41,39,43)(22,42,38,24)(26,33,44,30);(5,7)(6,10)(8,12)(9,11)(13,16,17,30,18,31,34,33)(14,20,22,39,23,40,38,21)(15,25,27,26,28,29,32,44)(19,35,24,43,37,36,42,41);(2,4)(7,12)(10,11)(13,16)(14,21)(15,26)(17,33)(18,31)(19,36)(20,38)(22,40)(23,39)(24,43)(25,27)(28,44)(29,32)(30,34)(35,37)(41,42)

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,e^b=ei,h^b=hi,d^c=de,g^c=gi,e^d=ei,g^d=gi,h^d=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, 384544);