#327129

order	512
nilpotency class	3
isoclinism	isoclinic class of #327117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

923 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 923 conjugacy classes of subgroups — too many to draw. Only its 148 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)(9,10,14,11)(12,15,13,16)(17,18,21,25)(19,22,26,30)(20,23,28,24)(27,29,32,31);(2,5)(6,8)(9,11,14,10)(12,15,13,16)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(1,3)(2,6)(4,7)(5,8)(10,15)(11,16)(12,13)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(3,7)(6,8)(10,11)(15,16)(19,26)(22,30)(27,32)(29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 234325);