#425811

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1340 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)(8,12)(9,11)(13,14)(15,25)(16,30)(17,34)(18,24)(19,23)(20,28)(21,33)(22,32)(26,43)(27,38)(29,40)(31,37)(35,41)(36,39)(42,44);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,15,19,36,18,29,17,28)(14,20,34,40,24,39,23,25)(16,31,35,27,33,44,32,26)(21,38,41,37,30,43,22,42);(5,8,9,7)(6,10,11,12)(13,16)(14,21)(15,26)(17,32)(18,33)(19,35)(20,37)(22,23)(24,30)(25,38)(27,29)(28,44)(31,36)(34,41)(39,42)(40,43);(2,4)(13,16)(14,22)(15,27)(17,35)(18,33)(19,32)(20,38)(21,23)(24,41)(25,37)(26,29)(28,44)(30,34)(31,36)(39,43)(40,42)

Polycyclic presentation:

2^9|b^2=ef,c^2=fi,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, 333007);