#7702518

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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	#45785	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4910 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)(9,10,12,13)(11,15,16,14)(17,18,21,23)(19,25,27,22)(20,24,28,30)(26,31,32,29);(1,3,5,4)(2,6,8,7)(10,13)(14,15)(17,19,21,27)(18,22,23,25)(20,26,28,32)(24,29,30,31);(9,10)(11,14)(12,13)(15,16)(18,23)(20,28)(22,25)(26,32);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,14,13,15)(17,20,21,28)(18,24,23,30)(19,26,27,32)(22,29,25,31);(10,13)(14,15)(18,23)(22,25)(24,30)(29,31);(1,4,5,3)(2,6,8,7)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,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, 7609714);