#271828

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

916 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,15)(8,11)(9,19)(10,12)(13,17)(14,16)(18,20)(21,22,25,28,26,27,34,35)(23,24,30,31,29,32,36,33);(1,2)(3,4)(5,7,10,17)(6,11,12,20)(8,9,18,16)(13,19,15,14)(21,23)(22,24)(25,30)(26,29)(27,32)(28,31)(33,35)(34,36);(2,4)(6,12)(11,20)(13,15)(14,19)(21,24,26,32)(22,23,27,29)(25,33,34,31)(28,36,35,30);(5,8,10,18)(6,13,12,15)(7,16,17,9)(11,14,20,19)(22,27)(23,29)(28,35)(30,36)

Polycyclic presentation:

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

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, 179024);