#119540

order	512
nilpotency class	3
isoclinism	isoclinic class of #119538
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

856 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 856 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,e^b=ei,g^b=gh,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26736);