#238076

order	512
nilpotency class	3
isoclinism	isoclinic class of #10390
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator, Frattini, cosocle	C₂⁵	C₂⁴
socle	C₂²	C₂×#393
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1033 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1033 conjugacy classes of subgroups — too many to draw. Only its 172 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) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=i,b^a=be,c^a=cf,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145272);