#385734

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1024 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 292930);