#247686

order	512
nilpotency class	3
isoclinism	isoclinic class of #247527
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1149 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154882);