#4976249

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 6291456
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3397 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=ghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326985, #72327007
maximal normal subgroup of	#72326985, #72327007
maximal quotient of	#72329174

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4883445);