#493308

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator	C₂²	C₂²×C₄×C₈
socle	C₂³	C₂³×C₈
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

712 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)(8,10)(9,12)(13,14,18,22,19,23,31,36,17,16,21,28,30,29,35,41)(15,24,26,37,27,38,40,44,25,20,32,33,39,34,42,43);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,24)(17,25)(18,26)(19,27)(21,32)(22,33)(23,34)(28,37)(29,38)(30,39)(31,40)(35,42)(36,43)(41,44);(7,12)(9,11)(13,16,18,28,19,29,31,41,17,14,21,22,30,23,35,36)(15,24,26,37,27,38,40,44,25,20,32,33,39,34,42,43);(6,10)(9,11)(15,25)(20,24)(26,32)(27,39)(33,37)(34,38)(40,42)(43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400504);