#7622783

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#3482	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1245 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,28,23,24,35,37,21,26,34,41,19,25,30,39)(20,27,32,42,33,36,45,46,31,40,44,48,29,38,43,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,21,23)(18,24,26,25)(20,29,31,33)(22,30,34,35)(27,36,40,38)(28,37,41,39)(32,43,44,45)(42,46,48,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,21,23)(18,25,26,24)(20,29,31,33)(22,30,34,35)(27,38,40,36)(28,39,41,37)(32,43,44,45)(42,47,48,46);(2,8)(6,9)(7,16)(14,15)(18,26)(24,25)(27,40)(28,41)(36,38)(37,39)(42,48)(46,47);(17,20,22,32,23,33,35,45,21,31,34,44,19,29,30,43)(18,27,28,42,24,36,37,46,26,40,41,48,25,38,39,47)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,e^2=g,g^2=h,h^2=i,b^a=bf,d^a=df,c^b=cfi

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529979);