#439344

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1480 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)(3,4)(5,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18,22,26)(19,23,20,24)(21,25,29,32)(27,30,28,31);(5,7,8,6)(9,11,13,16)(10,14,12,15)(17,19,21,28,22,20,29,27)(18,23,25,31,26,24,32,30);(6,7)(9,12,13,10)(11,15,16,14)(17,20)(18,24)(19,22)(21,28)(23,26)(25,31)(27,29)(30,32);(2,4)(6,7)(11,16)(14,15)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)

Polycyclic presentation:

2^9|a^2=efi,b^2=efg,c^2=ef,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=dei,g^c=gi

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, 346540);