#267970

order	512
nilpotency class	3
isoclinism	isoclinic class of #267947
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#61587	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1020 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175166);