#108607

order	512
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#382
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

530 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,c^b=ch,f^b=fhi,f^c=fhi

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, 15803);