#195259

order	512
nilpotency class	3
isoclinism	isoclinic class of #18284
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1186 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=h,d^2=hi,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dhi,f^a=fhi,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,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, 102455);