#396359

order	512
nilpotency class	3
isoclinism	isoclinic class of #395163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1455 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303555);