#455755

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

1011 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)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,20,19,24,17,23)(15,25,28,21,29,37,27,38)(16,30,34,40,33,35,32,22)(26,42,44,39,43,36,31,41);(5,7,8,11)(6,9,10,12)(14,20)(17,18)(21,25)(22,35)(23,24)(27,28)(30,40)(31,44)(32,34)(36,41)(37,38)(39,42);(6,10)(9,12)(13,15,19,29)(14,21,24,38)(16,31,33,44)(17,28,18,27)(20,25,23,37)(22,39,40,41)(26,34,43,32)(30,42,35,36);(2,4)(13,16)(14,22)(15,26)(17,34)(18,32)(19,33)(20,35)(21,36)(23,30)(24,40)(25,41)(27,44)(28,31)(29,43)(37,39)(38,42)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362951);