#7616792

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1421 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=fg,d^2=hi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313354, #72313540, #72313904
maximal normal subgroup of	#72313354, #72313540, #72313904

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7523988);