#368670

order	512
nilpotency class	3
isoclinism	isoclinic class of #24808
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1076 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=efhi,c^2=g,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fi,h^c=hi

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, 275866);