#8701747

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	#2679	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5415 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2)(3,7)(4,8)(5,6)(11,15)(13,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31);(2,6)(3,8)(9,10,12,14)(11,13,15,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32);(1,3)(2,4)(5,8)(6,7)(9,11)(10,13)(12,15)(14,16);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,20,23,18)(19,24,21,26)(22,25,30,28)(27,32,29,31);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,21,23,19)(18,24,20,26)(22,27,30,29)(25,32,28,31);(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)

Polycyclic presentation:

2^9|b^2=gh,d^2=gi,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72340594
maximal subgroup of	#72340548, #72340560, #72340574, …
maximal normal subgroup of	#72340548, #72340560, #72340574, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8608943);