#246196

order	512
nilpotency class	3
isoclinism	isoclinic class of #246159
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	C₂³×C₄	C₂⁴
prime radical	#2656	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1145 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)(7,12)(8,16)(9,11)(10,14)(13,15)(17,19)(18,20)(21,22,27,24)(23,30,25,34)(26,33,36,29)(28,35,32,31);(1,2)(3,4)(6,11)(7,9)(12,14)(13,16)(15,18)(19,20)(21,23,27,25)(22,28,24,32)(26,35,36,31)(29,34,33,30);(2,4)(5,7,10,9)(6,12,14,11)(8,15,17,18)(13,19,20,16)(21,24,27,22)(23,28,25,32)(26,29,36,33)(30,31,34,35);(5,8,10,17)(6,13,14,20)(7,15,9,18)(11,16,12,19)(21,25,27,23)(22,28,24,32)(26,31,36,35)(29,34,33,30)

Polycyclic presentation:

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

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, 153392);