#289428

order	512
nilpotency class	3
isoclinism	isoclinic class of #289395
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=ef,c^2=i,d^2=ei,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^b=ghi,d^c=dh,f^c=fi,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196624);