#7733252

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	#35395	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5346 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7640448);