#472006

order	512
nilpotency class	3
isoclinism	isoclinic class of #465214
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1354 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)(7,15,19,16)(8,12,22,28)(9,13)(10,14)(11,18,27,25)(17,29)(20,33,34,21)(23,24)(26,35)(30,32)(31,36);(1,2,3,4)(5,7,10,19)(6,11)(8,17,22,32)(9,18,24,25)(12,26)(13,15)(14,27)(16,23)(20,29)(21,31,33,35)(28,36)(30,34);(2,4)(5,6)(7,16)(8,20)(9,23)(10,14)(11,25)(12,21)(13,24)(15,19)(17,26)(18,27)(22,34)(28,33)(29,35)(30,31)(32,36);(5,8,9,21,10,22,24,33)(6,12,13,20,14,28,23,34)(7,17,18,31,19,32,25,35)(11,26,15,29,27,36,16,30)

Polycyclic presentation:

2^9|a^2=f,b^2=eghi,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,h^c=hi,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, 379202);