#7707732

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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	#51780	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5004 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)(11,12)(14,16)(17,18,21,24)(19,22,26,30)(20,27,28,23)(25,31,32,29);(2,5)(6,8)(9,10,13,15)(11,16,12,14)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31);(9,11)(10,14)(12,13)(15,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31);(10,15)(14,16)(20,28)(23,27)(25,32)(29,31);(9,12,13,11)(10,14,15,16)(18,24)(22,30)(23,27)(29,31);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7614928);