#130593

order	512
nilpotency class	4
isoclinism	isoclinic class of #130532
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	#2543	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2353 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

599 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 599 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

(2,6,7,8)(9,10)(11,16,18,24)(12,15,20,14)(13,17)(19,23)(21,22);(1,2)(3,6)(4,8)(5,7)(10,14)(11,18)(12,20)(15,17)(16,23)(22,24);(1,3,5,4)(2,6,7,8)(9,11,13,18)(12,21,20,19)(14,15)(16,24)

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=ei,f^c=fg,f^d=fi,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37789);