#4131222

order	512
nilpotency class	2
isoclinism	isoclinic class of #4122169
automorphism	Aut(G) order 25165824
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2098 conjugacy classes of subgroups; only the 686 normal subgroups are shown.

normal subgroups only

This group has 2098 conjugacy classes of subgroups — too many to draw. Only its 686 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=h,c^2=h,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#72267642, #72267659
Fitting subgroup of	#72267654
maximal subgroup of	#72267641, #72267654, #72268147, …
maximal normal subgroup of	#72267641, #72267654, #72268147, …
maximal quotient of	#72267648

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4038418);