#2953396

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

3186 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fgh,c^2=fghi,d^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300711, #72300749, #72300780
maximal normal subgroup of	#72300711, #72300749, #72300780

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2860592);