#7641779

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5044 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,26)(20,30)(21,24)(22,25)(23,28)(27,31)(29,32);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,30)(21,24)(22,25)(26,28)(27,32)(29,31);(10,13)(14,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,31)(25,26)(30,32);(9,11,12,16)(10,14,13,15)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,31,26,32);(1,3,4,7)(2,6,5,8)(19,28)(23,26)(27,29)(31,32);(17,21,22,20)(18,24,25,30)(19,29,28,27)(23,31,26,32)

Polycyclic presentation:

2^9|d^2=h,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280742, #72281045
maximal normal subgroup of	#72280742, #72281045
maximal quotient of	#72282088

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7548975);