#464747

order	512
nilpotency class	3
isoclinism	isoclinic class of #454982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1492 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)(7,16,18,11)(8,12)(9,13)(10,14)(15,23)(17,24,31,27)(19,26,33,25)(20,28)(21,29)(22,30)(32,35,34,36);(1,2,3,4)(5,7)(6,11)(8,17)(9,19)(10,18)(12,27)(13,25)(14,16)(15,26)(20,32)(21,31)(22,34)(23,33)(24,29)(28,36)(30,35);(2,4)(5,8,10,21)(6,12,14,29)(7,17,18,31)(9,22,23,20)(11,24,16,27)(13,30,15,28)(19,34,33,32)(25,36,26,35);(6,13,14,15)(11,25,16,26)(12,28,29,30)(24,35,27,36)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286018, #72286090, #72286106
maximal normal subgroup of	#72286018, #72286090, #72286106

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371943);