#427240

order	512
nilpotency class	3
isoclinism	isoclinic class of #417538
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1529 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)(7,11)(8,12)(9,13)(10,14)(15,25)(16,28)(17,24)(18,26)(19,27)(20,23)(21,29)(22,31)(30,34)(32,35)(33,36);(5,7)(6,11)(8,15)(9,20)(10,17)(12,25)(13,28)(14,24)(16,21)(18,32)(19,31)(22,27)(23,29)(26,34)(30,33)(35,36);(5,8,10,19)(6,12,14,27)(7,15,17,31)(9,18,21,33)(11,22,24,25)(13,26,29,36)(16,30,20,32)(23,34,28,35);(2,4)(5,9)(6,13)(7,16,17,20)(8,18)(10,21)(11,23,24,28)(12,26)(14,29)(15,30,31,32)(19,33)(22,34,25,35)(27,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284700, #72284731, #72284821, …
maximal normal subgroup of	#72284700, #72284731, #72284821, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334436);