#480698

order	512
nilpotency class	4
isoclinism	isoclinic class of #480670
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=ch,d^b=di,f^b=fg,d^c=di,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387894);