#196037

order	512
nilpotency class	3
isoclinism	isoclinic class of #184831
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6,9,11,10,7,8,12)(13,14)(15,19,25,32)(16,20)(17,21)(18,22)(23,30,38,42)(24,31,40,44)(26,33)(27,34)(28,35)(29,36)(37,41,39,43);(5,7,9,12,10,6,8,11)(13,15,18,25)(14,19)(16,23,27,38)(17,24,29,40)(20,30)(21,31)(22,32)(26,37,28,39)(33,41)(34,42)(35,43)(36,44);(2,4)(13,16,18,27)(14,20,22,34)(15,23,25,38)(17,28,29,26)(19,30,32,42)(21,35,36,33)(24,39,40,37)(31,43,44,41);(5,8,10,9)(6,11,7,12)(13,17,18,29)(14,21,22,36)(15,24,25,40)(16,26,27,28)(19,31,32,44)(20,33,34,35)(23,37,38,39)(30,41,42,43)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=di,e^b=eh,g^b=gh,d^c=dh

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, 103233);