#209092

order	512
nilpotency class	3
isoclinism	isoclinic class of #18346
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	#392	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

849 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)(9,10,14,13,15,18,27,26)(11,16,22,20,21,29,36,35)(12,17,25,23,24,30,38,37)(19,31,34,40,33,28,39,32);(1,2,4,6)(3,8,7,5)(9,11)(10,16)(12,19)(13,20)(14,22)(15,21)(17,28)(18,29)(23,32)(24,33)(25,34)(26,35)(27,36)(30,31)(37,40)(38,39);(9,12)(10,17)(11,19)(13,23)(14,25)(15,24)(16,28)(18,30)(20,32)(21,33)(22,34)(26,37)(27,38)(29,31)(35,40)(36,39);(2,6)(5,8)(9,13,14,18,15,26,27,10)(11,20,22,29,21,35,36,16)(12,23,25,30,24,37,38,17)(19,32,34,31,33,40,39,28)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116288);