#265128

order	512
nilpotency class	3
isoclinism	isoclinic class of #265099
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	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

853 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ei,c^2=i,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fhi,g^b=ghi,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172324);