#399121

order	512
nilpotency class	3
isoclinism	isoclinic class of #394746
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#397
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1357 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 306317);