#358347

order	512
nilpotency class	3
isoclinism	isoclinic class of #2081
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#397
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#397
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1462 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,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, 265543);