#427514

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1329 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)(7,10)(8,9)(11,12)(13,14,19,24)(15,20,26,36)(16,21,30,39)(17,23,18,22)(25,33,41,44)(27,34,28,35)(29,43,42,37)(31,38,32,40);(1,2,3,4)(13,15)(14,20)(16,29,30,42)(17,27)(18,28)(19,26)(21,37,39,43)(22,35)(23,34)(24,36)(25,31,41,32)(33,40,44,38);(6,9)(10,12)(13,16)(14,21)(15,25)(17,32)(18,31)(19,30)(20,33)(22,38)(23,40)(24,39)(26,41)(27,42)(28,29)(34,37)(35,43)(36,44);(2,4)(5,7,8,11)(6,10,9,12)(14,22)(15,26)(17,18)(20,34)(21,38)(23,24)(25,41)(31,32)(33,37)(35,36)(39,40)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=dei,e^c=ei,g^c=gi,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, 334710);