#5079945

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2972 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fghi,c^2=gh,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326968, #72326980, #72326992
maximal normal subgroup of	#72326968, #72326980, #72326992

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4987141);