#212999

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	#23798	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

824 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=fh,d^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ci,d^c=dh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120195);