#108275

order	512
nilpotency class	3
isoclinism	isoclinic class of #108262
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	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

743 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)(3,10,12,7)(4,6)(5,9)(8,13)(11,15,16,14)(17,18,21,26,23,28,22,27)(19,24,31,38,30,37,32,39)(20,25,34,41,33,40,35,42)(29,43,45,48,44,36,46,47);(2,6,9,8)(7,14,10,15)(17,19,22,32,23,30,21,31)(18,24,27,39,28,37,26,38)(20,29,35,46,33,44,34,45)(25,36,42,48,40,43,41,47);(1,3)(2,7)(4,11)(5,12)(6,14)(8,15)(9,10)(13,16)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,36)(26,41)(27,42)(28,40)(30,44)(31,45)(32,46)(37,43)(38,47)(39,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15471);