#3332694

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

3055 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gh,c^2=ghi,d^2=fghi,e^2=ghi,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	#72334161, #72334170, #72334239
maximal normal subgroup of	#72334161, #72334170, #72334239

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3239890);