#2623253

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

2756 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)(3,4)(5,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,20)(21,24)(22,23)(25,26,29,31)(27,32,28,30);(1,3)(2,6)(4,7)(5,8)(9,11,13,16)(10,14,15,12)(17,19)(18,22)(20,24)(21,23)(25,27)(26,30)(28,29)(31,32);(9,11)(10,12)(13,16)(14,15)(17,20)(18,23)(19,24)(21,22)(25,26)(27,32)(28,30)(29,31);(1,4,5,6)(2,3,7,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24)(25,28,29,27)(26,30,31,32);(1,3)(2,4)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,30)(28,32)(29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326570, #72326606, #72326611
maximal normal subgroup of	#72326570, #72326606, #72326611
maximal quotient of	#72328736

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2530449);