#7670083

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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

5391 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)(3,7,4,6)(9,10,13,12)(11,15,16,14)(17,18,21,24)(19,25,26,22)(20,28,30,23)(27,31,29,32);(1,3,5,4)(2,6,8,7)(9,11,13,16)(10,14,12,15)(17,19,21,26)(18,22,24,25)(20,27,30,29)(23,31,28,32);(10,12)(14,15)(17,19)(18,22)(20,29)(21,26)(23,32)(24,25)(27,30)(28,31);(1,3,5,4)(2,7,8,6)(9,12,13,10)(11,15,16,14)(19,26)(22,25)(27,29)(31,32);(17,20,21,30)(18,23,24,28)(19,27,26,29)(22,31,25,32);(1,4,5,3)(2,6,8,7)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312630, #72342307
maximal normal subgroup of	#72312630, #72342307

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577279);