#210175

order	512
nilpotency class	3
isoclinism	isoclinic class of #18930
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

939 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=ghi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 117371);