#428799

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1078 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335995);