#7625725

order	512
nilpotency class	2
isoclinism	isoclinic class of #62842
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	C₂⁵	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6918 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=g,e^2=h,f^2=g,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, 7532921);