#427380

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283460, #72283578, #72283596, …
maximal normal subgroup of	#72283460, #72283578, #72283596, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334576);