#363461

order	512
nilpotency class	3
isoclinism	isoclinic class of #184831
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=egi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270657);