#357847

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×#388
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2654 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:

(2,4)(5,6,10,14,11,15,9,25)(7,16,20,28,19,32,18,12)(8,21,24,13,23,30,26,29)(17,27,33,31,22,36,34,35);(1,2,3,4)(5,7,9,18,11,19,10,20)(6,12,14,16,15,28,25,32)(8,22,24,34,23,17,26,33)(13,27,21,35,29,36,30,31);(5,8)(6,13)(7,17)(9,26)(10,24)(11,23)(12,27)(14,30)(15,29)(16,31)(18,34)(19,22)(20,33)(21,25)(28,36)(32,35);(8,23)(13,29)(17,22)(21,30)(24,26)(27,36)(31,35)(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,g^a=gi,c^b=cg,g^b=gi,d^c=di,e^c=ei,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, 265043);