#400674

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1426 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:

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

Polycyclic presentation:

2^9|b^2=ef,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi

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, 307870);