#7780303

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

5096 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311104, #72311122, #72311201, …
maximal normal subgroup of	#72311104, #72311122, #72311201, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7687499);