#476984

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
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

1075 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)(7,10)(8,11)(9,12)(13,14,17,22,18,23,32,34)(15,19,25,35,26,36,31,21)(16,27,28,38,29,39,30,20)(24,37,40,43,41,44,42,33);(5,7)(6,9)(8,12)(10,11)(13,15,17,25,18,26,32,31)(14,19,22,35,23,36,34,21)(16,24,28,40,29,41,30,42)(20,33,27,37,38,43,39,44);(5,7)(6,10)(8,12)(9,11)(13,16,18,29)(14,20,23,38)(15,24,26,41)(17,30,32,28)(19,33,36,43)(21,37,35,44)(22,39,34,27)(25,42,31,40);(2,4)(7,12)(9,10)(13,15)(14,21)(16,24)(17,31)(18,26)(19,34)(20,37)(22,36)(23,35)(25,32)(27,33)(28,42)(29,41)(30,40)(38,44)(39,43)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384180);