#7628643

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
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	#35259	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5156 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)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,24)(21,29)(22,25)(26,30)(27,31)(28,32);(2,5)(6,8)(17,19)(18,23)(20,28)(21,26)(22,27)(24,32)(25,31)(29,30);(10,13)(14,16)(17,20,22,21)(18,24,25,29)(19,26,27,28)(23,30,31,32);(9,11,12,15)(10,14,13,16)(19,27)(23,31)(26,28)(30,32);(17,21,22,20)(18,24,25,29)(19,26,27,28)(23,32,31,30);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7535839);