#7701740

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
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	#47683	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4906 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dh,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, 7608936);