#3071226

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×#2671	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2961 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,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	#72280903, #72281051, #72281188
maximal normal subgroup of	#72280903, #72281051, #72281188

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2978422);