#214584

order	512
nilpotency class	3
isoclinism	isoclinic class of #163298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#35292	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1082 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ce,d^b=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121780);