#3995187

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2804 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fgi,c^2=fgh,d^2=ghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300704, #72300749, #72300774
maximal normal subgroup of	#72300704, #72300749, #72300774

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3902383);