#3071225

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2668	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3262 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:

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

Polycyclic presentation:

2^9|c^2=fh,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280883, #72281083, #72281188
maximal normal subgroup of	#72280883, #72281083, #72281188

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2978421);