#263358

order	512
nilpotency class	3
isoclinism	isoclinic class of #263351
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

914 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 144 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)(3,11)(4,10)(5,9)(6,8)(7,13)(12,16)(14,15)(18,22)(19,26)(20,23)(25,31)(28,29)(30,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,16)(9,11)(10,15)(17,18,23,31,21,25,20,22)(19,27,26,30,29,24,28,32);(2,8)(4,6)(7,15)(9,10)(11,16)(12,14)(17,19)(18,24)(20,28)(21,29)(22,30)(23,26)(25,27)(31,32);(1,4,5,6)(2,8,9,10)(3,12,13,14)(7,16,11,15)(17,18,23,31,21,25,20,22)(19,24,26,32,29,27,28,30)

Polycyclic presentation:

2^9|b^2=ei,d^2=eh,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=dhi,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305438, #72305449, #72305464
maximal normal subgroup of	#72305438, #72305449, #72305464

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170554);