#232170

order	512
nilpotency class	3
isoclinism	isoclinic class of #232169
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#23802	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 607 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)(3,8,11,7)(4,12,14,16)(5,9)(6,10)(13,15)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,3)(2,7)(4,13)(5,14)(6,11)(8,10)(9,12)(15,16);(1,4,13,3,6,14,5,11)(2,8,9,12,10,7,15,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(2,9,10,15)(7,16,8,12)(19,25)(22,28)(26,31)(29,32)

Polycyclic presentation:

2^9|a^2=g,c^2=efh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,c^b=cef,d^b=de,f^b=fh,g^b=gh,d^c=di,g^c=gh

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, 139366);