#272404

order	512
nilpotency class	3
isoclinism	isoclinic class of #272283
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

796 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=i,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=gi,f^c=fi,g^c=gi,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, 179600);