#209067

order	512
nilpotency class	3
isoclinism	isoclinic class of #18162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#120
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116263);