#347454

order	512
nilpotency class	3
isoclinism	isoclinic class of #347434
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#63045	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2287 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:

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

Polycyclic presentation:

2^9|a^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fhi,g^a=gh,e^b=eh,f^b=fi,e^c=ei,f^c=fhi,g^d=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268466, #72271994
maximal normal subgroup of	#72268466

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254650);