#194492

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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

1198 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)(7,14,15,17)(11,18,20,19)(21,22,25,30,26,28,24,29)(23,31,34,27,33,36,32,35);(1,2,3,4)(5,6)(7,11)(8,12)(9,16)(10,13)(14,19)(15,20)(17,18)(21,23)(22,27)(24,34)(25,32)(26,33)(28,35)(29,36)(30,31);(5,7)(6,11)(8,14)(9,17)(10,15)(12,19)(13,20)(16,18)(22,28)(24,25)(27,35)(32,34);(5,8,10,9)(6,12,13,16)(7,14,15,17)(11,19,20,18)(22,28)(27,35)(29,30)(31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101688);