#195626

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=hi,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, 102822);