#117121

order	512
nilpotency class	3
isoclinism	isoclinic class of #102045
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

798 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^c=dh,e^c=ehi,f^c=fh,g^c=gh

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, 24317);