#6465556

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3153	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,d^c=di

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, 6372752);