#455978

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1204 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:

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

Polycyclic presentation:

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

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, 363174);