#190363

order	512
nilpotency class	3
isoclinism	isoclinic class of #190208
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=i,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,g^b=gh,f^c=fi,g^c=gh

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, 97559);