#7711925

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619121);