#122965

order	512
nilpotency class	4
isoclinism	isoclinic class of #12087
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#442
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₈²	C₂³
socle, prime radical	C₂²	#1580
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1312 ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=ef,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=cfi,d^a=df,f^a=fh,c^b=ci,d^b=dfh,e^b=eh,f^b=fh,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286442, #72286483, #72286988, …
maximal normal subgroup of	#72286442, #72286483, #72286988, …
maximal quotient of	#72286990, #72323195

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30161);