#425442

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

1032 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)(5,6,8,10)(7,11,12,9)(13,14,18,22,19,23,32,17)(15,24,26,35,27,39,33,20)(16,21,30,37,31,38,44,29)(25,40,41,36,28,42,43,34);(13,15,19,27)(14,20,23,35)(16,28,31,25)(17,24,22,39)(18,33,32,26)(21,36,38,34)(29,42,37,40)(30,41,44,43);(5,7,8,12)(6,9,10,11)(13,16,18,30,19,31,32,44)(14,21,22,37,23,38,17,29)(15,25,26,41,27,28,33,43)(20,34,24,40,35,36,39,42);(2,4)(13,17,32,23,19,22,18,14)(15,20,33,39,27,35,26,24)(16,29,44,38,31,37,30,21)(25,34,43,42,28,36,41,40)

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=ef,d^2=ei,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, 332638);