#413611

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

966 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,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,f^c=fi

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, 320807);