#8818967

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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

6223 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249472, #72249519, #72249522
maximal normal subgroup of	#72249472, #72249519, #72249522

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8726163);