#495329

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

868 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,g^2=h,h^2=i,b^a=be,c^a=cf,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402525);