#246318

order	512
nilpotency class	3
isoclinism	isoclinic class of #246239
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	C₂×#538	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1171 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)(7,16)(8,13)(9,12)(10,19)(11,15)(14,17)(18,20)(21,22,26,29)(23,30,24,32)(25,34,35,28)(27,33,36,31);(1,2)(3,4)(5,7,11,17)(6,12,15,20)(8,9,10,18)(13,14,19,16)(21,23)(22,27)(24,26)(25,33)(28,32)(29,36)(30,34)(31,35);(2,4)(5,8,11,10)(6,13,15,19)(7,9,17,18)(12,14,20,16)(25,35)(28,34)(30,32)(31,33);(5,9,11,18)(6,14,15,16)(7,8,17,10)(12,13,20,19)(21,24,26,23)(22,27,29,36)(25,31,35,33)(28,32,34,30)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153514);