#8077084

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	C₂×H(3,𝔽₄)	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5047 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,20,22)(19,23,24,21);(9,10,13,11)(12,14,16,15)(17,19,20,24)(18,21,22,23);(2,5)(6,8)(10,11)(12,16);(3,7)(6,8)(12,16)(14,15);(1,3,4,7)(2,6,5,8)(9,11,13,10)(12,15,16,14);(1,2)(3,6)(4,5)(7,8)(9,12,13,16)(10,14,11,15)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72268094, #72268173
Fitting subgroup of	#72268094
maximal subgroup of	#72268094, #72268162
maximal normal subgroup of	#72268094, #72268162
maximal quotient of	#72268079, #72268147

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7984280);