#129850

order	512
nilpotency class	4
isoclinism	isoclinic class of #129837
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	C₂²×C₂²⋊C₄	C₂³
prime radical	#2617	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2273 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

681 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 681 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

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

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, 37046);