#7700056

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	#36465	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5501 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7607252);