#194091

order	512
nilpotency class	3
isoclinism	isoclinic class of #18712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1302 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fi,g^c=gi

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, 101287);