#6400617

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2676	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1767 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=dfi,e^c=ei,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6307813);