#7967778

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	#37674	C₂
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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7874974);