#245196

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
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1231 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1231 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) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=hi,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, 152392);