#7625698

order	512
nilpotency class	2
isoclinism	isoclinic class of #62842
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 44 points:

(1,2,3,4)(6,9)(10,12)(14,15)(18,31)(20,23)(21,24)(27,40)(29,41)(32,35)(38,44);(5,6)(7,10)(8,9)(11,12)(13,14,19,15)(16,20,28,23)(17,24,30,21)(18,22,31,25)(26,35,39,32)(27,33,40,36)(29,37,41,34)(38,43,44,42);(5,7,8,11)(6,10,9,12)(13,15,19,14)(16,20,28,23)(17,24,30,21)(18,25,31,22)(26,32,39,35)(27,33,40,36)(29,37,41,34)(38,42,44,43);(13,16)(14,20)(15,23)(17,26)(18,27)(19,28)(21,32)(22,33)(24,35)(25,36)(29,38)(30,39)(31,40)(34,42)(37,43)(41,44);(13,17,19,30)(14,21,15,24)(16,26,28,39)(18,29,31,41)(20,32,23,35)(22,34,25,37)(27,38,40,44)(33,42,36,43);(13,18)(14,22)(15,25)(16,27)(17,29)(19,31)(20,33)(21,34)(23,36)(24,37)(26,38)(28,40)(30,41)(32,42)(35,43)(39,44)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,e^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, 7532894);