#2526947

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2724 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)(3,8)(4,6)(5,7)(9,10,13,16)(11,14,12,15)(17,18,22,23)(19,24,21,20)(25,26)(27,31)(28,30)(29,32);(1,3,6,5)(2,7,4,8)(9,11,13,12)(10,14,16,15)(17,19,22,21)(18,20,23,24)(25,27,30,29)(26,31,28,32);(3,5)(7,8)(18,23)(19,21)(25,26,30,28)(27,32,29,31);(1,4,6,2)(3,7,5,8)(9,10)(11,14)(12,15)(13,16)(17,20,22,24)(18,19,23,21)(25,28,30,26)(27,31,29,32);(1,5,6,3)(2,7,4,8)(9,12,13,11)(10,15,16,14)(17,21,22,19)(18,24,23,20)(25,29,30,27)(26,31,28,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=fghi,c^2=fh,d^2=h,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326845, #72326846, #72326935
maximal normal subgroup of	#72326845, #72326846, #72326935

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2434143);