#7785383

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
prime radical	C₂×#538	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5558 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72267403, #72267439, #72267444, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7692579);