#202015

order	512
nilpotency class	3
isoclinism	isoclinic class of #201938
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60359	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2022 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)(3,7,8,5)(4,6)(9,10,14,18,15,19,13,29)(11,20,24,31,22,35,23,16)(12,25,28,17,26,33,27,32)(21,30,37,34,38,40,36,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16,19,31)(12,21)(13,23)(14,24)(15,22)(17,30,32,40)(18,20,29,35)(25,39,33,34)(26,38)(27,36)(28,37);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,21)(13,27)(14,28)(15,26)(16,30)(18,33)(19,32)(20,34)(22,38)(23,36)(24,37)(25,29)(31,40)(35,39);(11,22)(12,26)(16,31)(17,32)(20,35)(23,24)(25,33)(27,28)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235178, #72235200, #72235210, …
maximal normal subgroup of	#72235178, #72235200, #72235210, …
maximal quotient of	#72235820, #72262602

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109211);