#267527

order	512
nilpotency class	3
isoclinism	isoclinic class of #267467
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	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

841 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=h,f^2=h,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=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 174723);