#246860

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
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	#45976	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1227 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154056);