#511945

order	512
nilpotency class	4
isoclinism	isoclinic class of #511779
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#3496	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1113 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,c^b=ci,d^c=di,g^c=ghi,f^d=fh,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280964, #72281129, #72281265
maximal normal subgroup of	#72280964, #72281129, #72281265

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419141);