#355814

order	512
nilpotency class	3
isoclinism	isoclinic class of #3192
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,3)(10,13)(11,15)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,32)(30,31);(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(18,22)(23,31)(24,29)(25,32);(10,13)(14,16)(17,19,20,26)(18,23,22,31)(21,30,28,27)(24,32,29,25);(1,3,4,2)(9,11,12,15)(10,14,13,16)(18,22)(23,31)(24,29)(25,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=egi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cei,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263010);