#7514141

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514140
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#58990
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3062 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214082, #72226306, #72286053
maximal normal subgroup of	#72214082, #72286053
maximal quotient of	#72341354

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421337);