#195624

order	512
nilpotency class	3
isoclinism	isoclinic class of #18618
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	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1270 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)(5,6,8,12,10,13,9,14)(7,15,17,20,16,19,18,11)(21,22)(23,29)(24,26)(25,33)(27,31)(28,32)(30,36)(34,35);(1,2,3,4)(5,7)(6,11)(8,18)(9,17)(10,16)(12,19)(13,20)(14,15)(22,26)(27,35)(28,33)(29,36);(6,12)(8,9)(11,19)(13,14)(15,20)(17,18)(21,23,24,30)(22,27,26,35)(25,34,32,31)(28,36,33,29);(6,13)(7,16)(12,14)(17,18)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102820);