#2837641

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
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	#16249	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2882 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)(4,7)(5,6)(9,10)(11,14)(12,16)(13,15)(18,20)(19,23)(26,28)(27,32);(2,5)(3,4)(9,11,13,16)(10,14,15,12)(19,23)(22,24)(25,26)(27,30)(28,29)(31,32);(1,3,6,4)(2,7,5,8)(9,11)(10,12)(13,16)(14,15)(17,18,21,20)(19,22,23,24)(25,27)(26,30)(28,31)(29,32);(1,4,6,3)(2,7,5,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,22)(20,24)(21,23)(25,28,29,26)(27,31,32,30);(1,5,6,2)(3,8,4,7)(9,12)(10,11)(13,14)(15,16)(17,20,21,18)(19,24,23,22)(25,27)(26,31)(28,30)(29,32)

Polycyclic presentation:

2^9|b^2=ghi,c^2=fgh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232609, #72232627, #72232629
maximal normal subgroup of	#72232609, #72232627, #72232629
maximal quotient of	#72328440, #72328733

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2744837);