#7504291

order	512
nilpotency class	3
isoclinism	isoclinic class of #31282
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2684
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2684
prime radical	C₂×#3018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2253 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)(4,19)(5,10)(6,11)(7,12)(8,16)(9,21)(14,26)(15,18)(17,25)(20,24)(22,27)(23,28)(29,31)(30,32);(1,3,6,17,7,18,5,16)(2,8,10,15,12,25,11,13)(4,20,21,32,22,14,23,31)(9,26,19,30,28,24,27,29);(1,4)(2,9)(3,14)(5,21)(6,23)(7,22)(8,24)(10,27)(11,19)(12,28)(13,29)(15,30)(16,31)(17,32)(18,20)(25,26);(1,2)(3,15)(4,9)(5,10)(6,11)(7,12)(8,17)(13,18)(14,30)(16,25)(19,23)(20,29)(21,27)(22,28)(24,32)(26,31);(1,5,7,6)(2,10,12,11)(3,16,18,17)(4,21,22,23)(8,15,25,13)(9,27,28,19)(14,31,20,32)(24,30,26,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263012, #72267112, #72279299, …
maximal normal subgroup of	#72263012, #72267112, #72279299, …
maximal quotient of	#72262977

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7411487);