#195041

order	512
nilpotency class	3
isoclinism	isoclinic class of #18284
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂×#386
prime radical	#34633	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1348 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:

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

Polycyclic presentation:

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

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, 102237);