#3301987

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
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

3081 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=fgi,d^2=gi,e^2=fgi,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	#72232597, #72232610, #72232658
maximal normal subgroup of	#72232597, #72232610, #72232658

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3209183);