#413456

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

970 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=ef,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi

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, 320652);