#272065

order	512
nilpotency class	3
isoclinism	isoclinic class of #272027
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	#60597	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

904 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179261);