#4310652

order	512
nilpotency class	2
isoclinism	isoclinic class of #4308740
automorphism	Aut(G) order 67108864
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3160 conjugacy classes of subgroups; only the 690 normal subgroups are shown.

normal subgroups only

This group has 3160 conjugacy classes of subgroups — too many to draw. Only its 690 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4217848);