#93569

order	512
nilpotency class	4
isoclinism	isoclinic class of #93553
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1300
commutator	C₂×C₄²	C₄²
Frattini, cosocle	C₄×#117	C₂²
socle	C₂	#7037
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄² ◁ C₄×#117 ◁ G
LECS	G ▷ C₄×#117 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

294 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 62 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 — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,e^2=f,g^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339723, #72339741, #72339742
maximal normal subgroup of	#72339723, #72339741, #72339742

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 765);