#375180

order	512
nilpotency class	3
isoclinism	isoclinic class of #196002
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1558 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=fhi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^c=fi,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, 282376);