#209926

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

958 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)(21,22,26,23)(24,31,32,36)(25,30,35,28)(27,34,29,33);(6,11)(8,9)(12,18)(13,14)(15,17)(19,20)(21,23,26,22)(24,29,32,27)(25,28,35,30)(31,34,36,33);(5,7)(6,12)(8,17)(9,15)(10,16)(11,18)(13,19)(14,20)(21,24,26,32)(22,27,23,29)(25,34,35,33)(28,36,30,31);(1,3,4,2)(5,8,10,9)(6,13,14,11)(7,15,16,17)(12,18,20,19)(24,32)(27,29)(31,36)(33,34)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=gi,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, 117122);