#357696

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2658 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264892);