#7745610

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5736 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)(3,8)(4,6)(5,7)(9,10,14,16)(11,12,13,15)(17,18,21,20)(19,23,24,22);(1,3,5,4)(2,6,7,8)(9,11,14,13)(10,15,16,12)(17,19,21,24)(18,22,20,23);(3,4)(6,8)(10,16)(11,13);(9,11,14,13)(10,12,16,15);(1,2)(3,8)(4,6)(5,7)(9,12,14,15)(10,11,16,13);(1,4,5,3)(2,6,7,8)(9,13,14,11)(10,15,16,12)(17,20,21,18)(19,23,24,22)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303495, #72303497, #72306585
maximal normal subgroup of	#72303495, #72303497, #72306585
maximal quotient of	#72303502, #72303520

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7652806);