#195275

order	512
nilpotency class	3
isoclinism	isoclinic class of #18414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 102471);