#497117

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	#2397	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

680 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=ce,d^b=di,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404313);