#422555

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1160 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=e,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329751);