#415907

order	512
nilpotency class	3
isoclinism	isoclinic class of #394192
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×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

1550 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)(7,11)(8,9)(10,12)(13,14,19,24)(15,25,27,36)(16,30,32,38)(17,23,18,22)(20,29,35,28)(21,33,37,31)(26,40,41,34)(39,43,44,42);(1,2)(3,4)(6,9)(10,11)(13,15,17,28,19,27,18,29)(14,20,22,25,24,35,23,36)(16,26,31,42,32,41,33,43)(21,34,30,39,37,40,38,44);(2,4)(13,16)(14,21)(15,26)(17,31)(18,33)(19,32)(20,34)(22,30)(23,38)(24,37)(25,39)(27,41)(28,42)(29,43)(35,40)(36,44);(5,7,8,12)(6,10,9,11)(15,27)(20,35)(25,36)(26,41)(28,29)(34,40)(39,44)(42,43)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di,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, 323103);