#501125

order	512
nilpotency class	4
isoclinism	isoclinic class of #501120
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×Q₈	C₂⁴
prime radical	#24418	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1106 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408321);