#271812

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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

944 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:

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

Polycyclic presentation:

2^9|a^2=fh,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, 179008);