#7627614

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	C₂×#3544	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^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, 7534810);