#6370894

order	512
nilpotency class	3
isoclinism	isoclinic class of #6370843
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

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×#381
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #3428 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4263 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)(3,12,15,7)(4,9)(5,11)(6,10)(8,14,21,28)(13,25,26,19)(16,22)(17,23)(18,24)(20,27,30,32)(29,31);(1,3)(2,7)(4,13)(5,14)(6,15)(8,11)(9,19)(10,12)(16,26)(17,27)(18,28)(20,23)(21,24)(22,25)(29,32)(30,31);(1,3)(2,8,10,21)(4,13)(5,14)(6,15)(7,11,12,24)(9,20,22,30)(16,26)(17,27)(18,28)(19,23,25,31)(29,32);(1,4)(2,9)(3,13)(5,17)(6,16)(7,19)(8,20)(10,22)(11,23)(12,25)(14,27)(15,26)(18,29)(21,30)(24,31)(28,32);(4,16)(9,22)(13,26)(17,29)(19,25)(20,30)(23,31)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307021, #72307042, #72319030, …
maximal normal subgroup of	#72307021, #72307042, #72319030, …
maximal quotient of	#72313302

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278090);