#213798

order	512
nilpotency class	3
isoclinism	isoclinic class of #213798
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#35537	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1035 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1035 conjugacy classes of subgroups — too many to draw. Only its 194 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)(3,10,12,6)(4,9)(5,8)(7,11,16,13)(14,15)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(1,3)(2,6)(4,13)(5,12)(7,15)(8,10)(9,16)(11,14)(19,25)(22,28)(26,31)(29,32);(1,3)(2,7)(4,13)(5,12)(6,9)(8,16)(10,15)(11,14)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(1,3)(2,6)(4,11)(5,12)(7,9)(8,10)(13,14)(15,16)(18,23)(19,25)(24,30)(26,31)

Polycyclic presentation:

2^9|a^2=g,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cei,f^b=fh,g^b=gh,d^c=di,f^c=fh,g^c=gh,g^d=gh

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, 120994);