#395699

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1855 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302895);