#3071232

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×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2836 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)(17,18,22,25)(19,20,27,23)(21,24,31,32)(26,28,29,30);(6,8)(10,11)(14,15)(17,19,22,27)(18,23,25,20)(21,29,31,26)(24,28,32,30);(5,6,7,8)(9,10,13,11)(12,14,16,15)(17,18)(19,23)(20,27)(21,24)(22,25)(26,30)(28,29)(31,32);(1,2)(3,4)(9,11,13,10)(12,15,16,14)(17,20,22,23)(18,19,25,27)(21,30,31,28)(24,29,32,26);(9,12)(10,14)(11,15)(13,16)(17,21,22,31)(18,24,25,32)(19,26,27,29)(20,28,23,30)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=h,d^2=g,e^2=f,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	#72281188, #72281269, #72281360
maximal normal subgroup of	#72281188, #72281269, #72281360

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2978428);