#415613

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1702 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322809);