#124546

order	512
nilpotency class	4
isoclinism	isoclinic class of #11915
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1533
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#376	C₂³
socle	C₂	#7595
prime radical	C₂⁴	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

381 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 84 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,d^2=gi,e^2=h,f^2=i,b^a=bd,c^a=cg,d^a=df,g^a=gi,d^b=dg,e^b=efg,f^b=fi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31742);