#302008

order	512
nilpotency class	3
isoclinism	isoclinic class of #301993
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3003	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1002 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fh,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=di,g^c=ghi

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, 209204);