#272630

order	512
nilpotency class	3
isoclinism	isoclinic class of #272539
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	#18537	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

800 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:

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

Polycyclic presentation:

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

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, 179826);