#355051

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

854 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)(7,19)(8,23,27,32)(9,15)(10,17)(11,16)(12,18)(13,21)(14,26,31,25)(20,30,35,28)(22,29)(24,36,34,33);(1,2,4,3)(5,7,11,9,12,22,10,21)(6,13,17,29,18,15,16,19)(8,20,26,24,27,35,25,34)(14,28,32,36,31,30,23,33);(5,8,10,25,12,27,11,26)(6,14,16,23,18,31,17,32)(7,20,21,34,22,35,9,24)(13,28,19,33,15,30,29,36);(1,3,4,2)(5,9,11,22,12,21,10,7)(6,15,16,29,18,13,17,19)(8,24,26,35,27,34,25,20)(14,30,23,36,31,28,32,33)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=efh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=di,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240734, #72307323
maximal normal subgroup of	#72240734, #72307323

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262247);