#209104

order	512
nilpotency class	3
isoclinism	isoclinic class of #18316
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

854 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)(9,10,14,13,15,18,28,27)(11,16,22,20,21,30,38,37)(12,17,25,26,24,31,39,23)(19,32,35,33,34,29,40,36);(1,2,4,6)(3,8,7,5)(9,11)(10,16)(12,19)(13,20)(14,22)(15,21)(17,29)(18,30)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(31,32)(39,40);(9,12)(10,17)(11,19)(13,26)(14,25)(15,24)(16,29)(18,31)(20,36)(21,34)(22,35)(23,27)(28,39)(30,32)(33,37)(38,40);(2,6)(5,8)(9,13,14,18,15,27,28,10)(11,20,22,30,21,37,38,16)(12,23,25,17,24,26,39,31)(19,33,35,29,34,36,40,32)

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,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116300);