#245164

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	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1117 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1117 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=hi,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, 152360);