#265338

order	512
nilpotency class	3
isoclinism	isoclinic class of #265335
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62253	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

900 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172534);