#8097746

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6997 conjugacy classes of subgroups; only the 2928 normal subgroups are shown.

normal subgroups only

This group has 6997 conjugacy classes of subgroups — too many to draw. Only its 2928 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) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=i,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72268338, #72271407
Fitting subgroup of	#72268338
maximal subgroup of	#72268338, #72271403, #72306585
maximal normal subgroup of	#72268338, #72271403, #72306585
maximal quotient of	#72268378, #72306597

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004942);