#355588

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
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×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1258 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)(9,10)(11,13)(12,15)(14,16)(17,18,21,25,20,24,29,23)(19,26,28,32,27,22,30,31);(5,6,7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,22)(20,27)(21,28)(23,32)(24,26)(25,31)(29,30);(1,2,3,4)(10,11)(14,15)(18,23)(19,27)(21,29)(22,31)(24,25)(26,32);(9,12,13,16)(10,14,11,15)(18,24)(19,27)(23,25)(28,30)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=efg,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=ce,d^b=di,f^c=fi,g^c=gi

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, 262784);