#209920

order	512
nilpotency class	3
isoclinism	isoclinic class of #18712
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	C₂×#127	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

936 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)(5,6,10,14)(7,12,16,20)(8,13,9,11)(15,18,17,19)(23,29)(24,31)(26,34)(27,32)(30,33)(35,36);(6,11)(8,9)(12,18)(13,14)(15,17)(19,20)(21,22,25,28)(23,26,30,35)(24,32,31,27)(29,36,33,34);(5,7,10,16)(6,12,14,20)(8,17,9,15)(11,18,13,19)(21,23)(22,26)(24,33)(25,30)(27,36)(28,35)(29,31)(32,34);(1,3,4,2)(5,8,10,9)(6,13,14,11)(7,15,16,17)(12,18,20,19)(23,30)(26,35)(29,33)(34,36)

Polycyclic presentation:

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

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, 117116);