#214031

order	512
nilpotency class	3
isoclinism	isoclinic class of #214020
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1039 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318396
maximal normal subgroup of	#72318396

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121227);