#125594

order	512
nilpotency class	4
isoclinism	isoclinic class of #125496
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

294 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,9,6,10,4,8)(3,7,11,14,12,15,13,16)(17,18)(19,29,31,25)(20,34,36,24)(21,27)(22,38)(23,28)(26,37,42,35)(30,39)(32,45,33,41)(40,47)(43,48)(44,46);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,14)(10,15)(18,24,28,34)(20,35)(21,22)(25,39,29,43)(26,27,42,38)(30,46)(32,33)(36,37)(40,41,44,45)(47,48);(1,4,6,5)(2,8,10,9)(3,11,12,13)(7,14,15,16)(17,19,23,31)(18,25,28,29)(20,30,36,48)(21,32,22,33)(24,39,34,43)(26,40,42,44)(27,41,38,45)(35,46,37,47)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32790);