#7649164

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

6004 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)(9,10,12,14)(11,15,16,13)(17,18,22,23)(19,25,27,21)(20,24,30,31)(26,32,28,29);(2,6)(3,8)(9,10,12,14)(11,13,16,15)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,30)(31,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(18,23)(21,25)(24,31)(29,32);(1,4)(2,3)(5,7)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,21)(20,28)(22,27)(23,25)(24,32)(26,30)(29,31);(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28);(17,21)(18,19)(20,29)(22,25)(23,27)(24,26)(28,31)(30,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262098, #72262117, #72262211, …
maximal normal subgroup of	#72262098, #72262117, #72262211, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7556360);