#209181

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
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

918 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,4,3)(6,11)(7,12)(8,9)(13,14,18,22)(15,23,26,37)(16,20,27,35)(17,29,32,21)(19,30,34,25)(24,40,41,39)(28,38,31,36)(33,43,44,42);(5,6,10,7)(8,12,9,11)(13,15)(14,19)(16,24)(17,30)(18,26)(20,33)(21,37)(22,34)(23,29)(25,32)(27,41)(28,43)(31,42)(35,44)(36,39)(38,40);(5,7,10,6)(8,12,9,11)(13,16,18,27)(14,20,22,35)(15,24,26,41)(17,31,32,28)(19,33,34,44)(21,38,29,36)(23,39,37,40)(25,43,30,42);(1,3,4,2)(6,7)(11,12)(16,27)(20,35)(24,41)(28,31)(33,44)(36,38)(39,40)(42,43)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116377);