#6371444

order	512
nilpotency class	3
isoclinism	isoclinic class of #60388
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#120
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂²×#386
prime radical	#62812	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6063 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,f^2=i,b^a=bf,c^a=cg,g^a=gi,f^b=fi,g^b=gi,h^b=hi,f^c=fi,h^c=hi,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215901, #72219684, #72219687, …
maximal normal subgroup of	#72215901, #72219684, #72219687, …
maximal quotient of	#72303339

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278640);