#368103

order	512
nilpotency class	3
isoclinism	isoclinic class of #23872
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1603 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^b=fi,g^b=gi,e^c=ei,f^c=fi,f^d=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, 275299);