#209615

order	512
nilpotency class	3
isoclinism	isoclinic class of #19014
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

881 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)(6,10)(8,15)(11,19)(12,16)(13,17)(18,20)(21,22,25,28)(23,29,30,36)(24,32,33,27)(26,34,35,31);(5,6)(7,11)(8,16)(9,12)(10,15)(13,18)(14,20)(17,19)(24,33)(27,32)(29,36)(31,34);(5,7,9,14)(6,11,12,20)(8,17,15,13)(10,18,16,19)(21,23)(22,26)(24,34)(25,30)(27,36)(28,35)(29,32)(31,33);(1,3,4,2)(6,12)(10,16)(11,20)(18,19)

Polycyclic presentation:

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

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, 116811);