#6371521

order	512
nilpotency class	3
isoclinism	isoclinic class of #6371520
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#386
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3585	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6001 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306939, #72307018, #72307052, …
maximal normal subgroup of	#72306939, #72307018, #72307052, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278717);