#7627418

order	512
nilpotency class	2
isoclinism	isoclinic class of #63041
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6698 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534614);