#3069068

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3117 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)(10,13)(11,15)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31);(6,8)(10,13)(14,16)(17,19,23,20)(18,24,21,25)(22,28,30,27)(26,32,29,31);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31);(1,2)(3,4)(9,10)(11,14)(12,13)(15,16)(17,21,23,18)(19,24,20,25)(22,26,30,29)(27,32,28,31);(9,11)(10,14)(12,15)(13,16)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,d^2=f,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	#72266886, #72267407
maximal normal subgroup of	#72266886, #72267407
maximal quotient of	#72267035

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2976264);