#108370

order	512
nilpotency class	3
isoclinism	isoclinic class of #108352
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

724 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:

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

Polycyclic presentation:

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

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, 15566);