#493192

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₁₆	C₂³
commutator	C₂²	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂³×C₈
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

706 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)(7,10)(8,9)(11,12)(13,14,17,22,18,23,30,36,19,16,21,28,31,29,35,41)(15,20,26,33,27,34,40,43,25,24,32,37,39,38,42,44);(1,2,3,4)(6,9)(10,12)(13,15)(14,20)(16,24)(17,26)(18,27)(19,25)(21,32)(22,33)(23,34)(28,37)(29,38)(30,40)(31,39)(35,42)(36,43)(41,44);(13,16,17,28,18,29,30,41,19,14,21,22,31,23,35,36)(15,24,26,37,27,38,40,44,25,20,32,33,39,34,42,43);(5,7,8,11)(6,10,9,12)(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,d^2=e,f^2=h,h^2=i,b^a=be,d^b=di

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400388);