#511836

order	512
nilpotency class	4
isoclinism	isoclinic class of #511835
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32898
commutator, Frattini, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2361 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=ci,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,g^d=gi,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72337673, #72340644
maximal normal subgroup of	#72337673, #72340644
maximal quotient of	#72337681, #72340694

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419032);