#2516755

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

2920 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=fg,e^2=fhi,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	#72326692, #72326711, #72326730, …
maximal normal subgroup of	#72326692, #72326711, #72326730, …
maximal quotient of	#72329054

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2423951);