#6448994

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	#35328	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356190);