#431449

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1103 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=ef,d^2=efi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^d=hi

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, 338645);