#190012

order	512
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#381
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1242 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97208);