#265412

order	512
nilpotency class	3
isoclinism	isoclinic class of #265399
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#18231	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

910 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:

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

Polycyclic presentation:

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

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, 172608);