#6373743

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#3150	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ci,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343748, #72343795
maximal normal subgroup of	#72343748, #72343795

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280939);