#504173

order	512
nilpotency class	4
isoclinism	isoclinic class of #501868
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1771 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322593
maximal normal subgroup of	#72322593

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411369);