#254294

order	512
nilpotency class	3
isoclinism	isoclinic class of #254275
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#120	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1037 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,f^d=fh

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, 161490);