#7669255

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
prime radical	#34962	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=i,e^2=g,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7576451);