#114859

order	512
nilpotency class	3
isoclinism	isoclinic class of #114856
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

893 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 893 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,e^c=eh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240217, #72240220, #72281054, …
maximal normal subgroup of	#72240217, #72240220, #72281054, …
maximal quotient of	#72240281, #72260929

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22055);