#374436

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1702 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,f^c=fi,h^c=hi

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, 281632);