#318709

order	512
nilpotency class	3
isoclinism	isoclinic class of #262675
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

574 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=egi,c^2=hi,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,f^c=fh,g^c=gi

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, 225905);