#108012

order	512
nilpotency class	3
isoclinism	isoclinic class of #108006
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,9,6,10,16,8)(3,11,12,15,13,14,7,4)(17,18,21,28,23,27,22,26)(19,29,31,24,32,39,33,40)(20,34,35,41,36,25,37,42)(30,43,44,47,45,48,46,38);(1,3,5,12,6,13,16,7)(2,4,9,11,10,15,8,14)(17,19,22,33,23,32,21,31)(18,24,28,39,27,40,26,29)(20,30,35,44,36,45,37,46)(25,38,41,48,34,47,42,43);(1,4)(2,7)(3,9)(5,11)(6,15)(8,13)(10,12)(14,16)(17,20)(18,25)(19,30)(21,37)(22,35)(23,36)(24,38)(26,42)(27,34)(28,41)(29,43)(31,46)(32,45)(33,44)(39,48)(40,47)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^b=ehi,d^c=dhi,e^c=ei,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15208);