#396716

order	512
nilpotency class	3
isoclinism	isoclinic class of #396715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1184 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)(5,6,11,16)(7,17,19,12)(8,21,24,32)(9,15,10,14)(13,26,30,23)(18,29,34,36)(20,28,25,31)(22,27,35,33);(5,7,9,25,11,19,10,20)(6,12,14,31,16,17,15,28)(8,22,23,34,24,35,26,18)(13,29,21,33,30,36,32,27);(5,8)(6,13)(7,18,19,34)(9,26)(10,23)(11,24)(12,27,17,33)(14,32)(15,21)(16,30)(20,22,25,35)(28,29,31,36);(2,4)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303912);