#6449175

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35328	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3374 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356371);