#209712

order	512
nilpotency class	3
isoclinism	isoclinic class of #18682
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

886 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=eh,f^b=fi,d^c=dh,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, 116908);