#471533

order	512
nilpotency class	3
isoclinism	isoclinic class of #464719
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1041 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378729);