#281712

order	512
nilpotency class	3
isoclinism	isoclinic class of #281659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

775 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 188908);