#6371071

order	512
nilpotency class	3
isoclinism	isoclinic class of #6370843
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#117
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂²×#383
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1935 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)(3,13,17,21)(4,10)(5,8)(6,11)(7,12)(9,16,24,15)(14,27,28,31)(18,25)(19,22)(20,26)(23,30,32,29);(2,8,12,11)(9,21,24,13)(10,22,25,26)(23,31,32,27);(1,3,5,15,7,17,6,16)(2,9,11,13,12,24,8,21)(4,14,19,29,18,28,20,30)(10,23,26,27,25,32,22,31);(1,4)(2,10)(3,14)(5,19)(6,20)(7,18)(8,22)(9,23)(11,26)(12,25)(13,27)(15,29)(16,30)(17,28)(21,31)(24,32);(4,18)(10,25)(14,28)(19,20)(22,26)(23,32)(27,31)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306939, #72307024, #72307055, …
maximal normal subgroup of	#72306939, #72307024, #72307055, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278267);