#245161

order	512
nilpotency class	3
isoclinism	isoclinic class of #245147
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1218 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1218 conjugacy classes of subgroups — too many to draw. Only its 156 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)(17,18,22,20)(19,25,27,32)(21,29,31,24)(23,30,26,28);(4,13)(6,15)(7,12)(10,14)(17,18,22,20)(19,23,27,26)(21,29,31,24)(25,28,32,30);(1,3,5,11)(2,6,8,15)(4,14,13,10)(7,16,12,9)(17,19,22,27)(18,23,20,26)(21,30,31,28)(24,32,29,25);(17,20,22,18)(19,26,27,23)(21,24,31,29)(25,30,32,28)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152357);