#414956

order	512
nilpotency class	3
isoclinism	isoclinic class of #393900
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1266 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)(8,9)(10,12)(13,14)(15,25)(16,30)(17,23)(18,22)(19,24)(20,27)(21,33)(26,42)(28,37)(29,36)(31,41)(32,40)(34,39)(35,44)(38,43);(2,4)(13,15,19,29)(14,20,24,37)(16,31,34,43)(17,28,18,27)(21,38,40,41)(22,36,23,25)(26,32,44,33)(30,42,39,35);(6,9)(10,11)(13,16,19,34)(14,21,24,40)(15,26,29,44)(17,33,18,32)(20,35,37,42)(22,30,23,39)(25,41,36,38)(27,31,28,43);(5,7,8,12)(6,10,9,11)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322152);