#7625769

order	512
nilpotency class	2
isoclinism	isoclinic class of #62842
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#3499	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

10582 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,f^a=fh,e^b=eh,d^c=dh

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532965);