#421057

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3013	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1872 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:

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

Polycyclic presentation:

2^9|b^2=ef,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei,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, 328253);