#263317

order	512
nilpotency class	3
isoclinism	isoclinic class of #263311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18186	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

858 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 858 conjugacy classes of subgroups — too many to draw. Only its 146 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)(3,6)(4,7)(5,9)(8,13)(10,14)(11,16)(12,15)(18,23)(19,26)(20,29)(21,25)(27,28)(31,32);(1,3,5,11)(2,6,9,16)(4,12,13,10)(7,15,8,14)(17,18,21,23,22,20,25,29)(19,24,27,31,28,30,26,32);(2,7)(4,13)(6,14)(8,9)(10,12)(15,16)(17,19)(18,24)(20,30)(21,27)(22,28)(23,31)(25,26)(29,32);(17,20,21,29,22,18,25,23)(19,24,27,31,28,30,26,32)

Polycyclic presentation:

2^9|b^2=ehi,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,d^c=dh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305430, #72305453, #72305464
maximal normal subgroup of	#72305430, #72305453, #72305464

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170513);