#392240

order	512
nilpotency class	3
isoclinism	isoclinic class of #392239
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14681
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3465 conjugacy classes of subgroups; only the 432 normal subgroups are shown.

normal subgroups only

This group has 3465 conjugacy classes of subgroups — too many to draw. Only its 432 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^b=hi,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299436);