#331273

order	512
nilpotency class	3
isoclinism	isoclinic class of #331233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

707 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 707 conjugacy classes of subgroups — too many to draw. Only its 162 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=h,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=dh,d^c=de,f^c=fh,g^c=gi

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, 238469);