#186847

order	512
nilpotency class	3
isoclinism	isoclinic class of #186842
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62824	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2342 conjugacy classes of subgroups; only the 310 normal subgroups are shown.

normal subgroups only

This group has 2342 conjugacy classes of subgroups — too many to draw. Only its 310 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) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,d^c=dh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94043);