#2568390

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	#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3589 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=gi,c^2=ghi,d^2=fgh,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	#72336774, #72336800, #72336814
maximal normal subgroup of	#72336774, #72336800, #72336814

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2475586);