#428239

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

1010 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)(7,10)(8,9)(11,12)(13,14,18,23,19,24,33,35)(15,25,27,22,28,39,17,20)(16,21,31,37,32,38,43,44)(26,40,41,36,29,42,30,34);(13,15,19,28)(14,20,24,22)(16,29,32,26)(17,33,27,18)(21,36,38,34)(23,39,35,25)(30,31,41,43)(37,40,44,42);(6,9)(10,12)(13,16,18,31,19,32,33,43)(14,21,23,37,24,38,35,44)(15,26,27,41,28,29,17,30)(20,34,25,40,22,36,39,42);(2,4)(5,7,8,11)(6,10,9,12)(13,17,18,15,19,27,33,28)(14,22,23,39,24,20,35,25)(16,30,31,26,32,41,43,29)(21,36,37,42,38,34,44,40)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=e,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,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, 335435);