#245181

order	512
nilpotency class	3
isoclinism	isoclinic class of #245168
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#36721	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1340 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1340 conjugacy classes of subgroups — too many to draw. Only its 152 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)(3,9)(4,12)(5,8)(6,10)(7,13)(11,16)(14,15)(18,22)(19,25)(21,27)(23,31)(26,29)(30,32);(4,13)(6,15)(7,12)(10,14)(17,18,20,22)(19,23,26,30)(21,28,27,24)(25,31,29,32);(1,3,5,11)(2,6,8,15)(4,14,13,10)(7,16,12,9)(17,19,20,26)(18,23,22,30)(21,29,27,25)(24,31,28,32);(21,27)(24,28)(25,29)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152377);