#7198641

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021351
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23798	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1667 conjugacy classes of subgroups; only the 508 normal subgroups are shown.

normal subgroups only

This group has 1667 conjugacy classes of subgroups — too many to draw. Only its 508 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 — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=ghi,d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7105837);