#267875

order	512
nilpotency class	3
isoclinism	isoclinic class of #267851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#32911	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

976 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175071);