#6448969

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 6356165);