#7625219

order	512
nilpotency class	2
isoclinism	isoclinic class of H(7,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄² · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂	C₂⁴×C₄²
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

12594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(5,6,8,7)(9,10)(11,16)(12,17)(13,25)(14,15)(18,26)(19,29)(20,34)(21,27)(22,31)(23,37)(24,28)(30,35)(32,38)(33,40)(36,39);(1,2,3,4)(5,7,8,6)(10,15)(16,27)(17,28)(18,25)(29,31)(30,34)(32,37)(39,40);(9,11,14,21)(10,16,15,27)(12,22,24,19)(13,20,26,35)(17,31,28,29)(18,30,25,34)(23,36,38,33)(32,40,37,39);(9,12,14,24)(10,17,15,28)(11,19,21,22)(13,23,26,38)(16,29,27,31)(18,32,25,37)(20,33,35,36)(30,39,34,40);(9,10)(11,16)(12,17)(13,18)(14,15)(19,29)(20,30)(21,27)(22,31)(23,32)(24,28)(25,26)(33,39)(34,35)(36,40)(37,38);(9,13)(10,18)(11,20)(12,23)(14,26)(15,25)(16,30)(17,32)(19,33)(21,35)(22,36)(24,38)(27,34)(28,37)(29,39)(31,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=g,b^a=bg,f^a=fg,e^b=eg,d^c=dg

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532415);