#416926

order	512
nilpotency class	3
isoclinism	isoclinic class of #394189
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

1080 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)(13,14,19,23,18,22,17,24)(15,25,34,36,28,37,27,20)(16,29,31,21,33,40,32,39)(26,38,42,41,30,35,43,44);(1,2)(3,4)(6,9)(10,11)(13,15,19,34,18,28,17,27)(14,20,24,37,22,36,23,25)(16,30,32,42,33,26,31,43)(21,38,40,41,39,35,29,44);(2,4)(13,16)(14,21)(15,26)(17,31)(18,33)(19,32)(20,35)(22,39)(23,29)(24,40)(25,41)(27,42)(28,30)(34,43)(36,38)(37,44);(5,7,8,12)(6,10,9,11)(14,22)(20,36)(21,39)(23,24)(25,37)(29,40)(35,38)(41,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=eh,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 324122);