#108371

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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=hi,g^2=hi,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, 15567);