#361920

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

1346 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=e,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, 269116);