#190579

order	512
nilpotency class	3
isoclinism	isoclinic class of #184831
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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 ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

992 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=e,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=dh,d^c=di,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, 97775);