#427375

order	512
nilpotency class	3
isoclinism	isoclinic class of #427374
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
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

1399 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)(7,11)(8,12)(9,20,22,13)(10,14)(15,26)(16,24,21,29)(17,25)(18,32,34,27)(19,28)(23,31)(30,36,33,35);(1,2)(3,4)(5,7)(6,11)(8,15)(9,21)(10,17)(12,26)(13,29)(14,25)(16,22)(18,33)(19,31)(20,24)(23,28)(27,35)(30,34)(32,36);(2,4)(5,8,10,19)(6,12,14,28)(7,15,17,31)(9,18,22,34)(11,23,25,26)(13,27,20,32)(16,30,21,33)(24,35,29,36);(5,9)(6,13)(7,16,17,21)(8,18)(10,22)(11,24,25,29)(12,27)(14,20)(15,30,31,33)(19,34)(23,35,26,36)(28,32)

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=f,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,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	#72283460, #72283700, #72283734, …
maximal normal subgroup of	#72283460, #72283700, #72283734, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334571);