#504128

order	512
nilpotency class	4
isoclinism	isoclinic class of #504125
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

542 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322971
maximal normal subgroup of	#72322971

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411324);