#356396

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

1122 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ef,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci,d^b=de,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263592);