#352835

order	512
nilpotency class	3
isoclinism	isoclinic class of #352122
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

738 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 202 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=fi,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,d^b=dhi,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260031);