#392546

order	512
nilpotency class	3
isoclinism	isoclinic class of #188764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

778 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240727, #72240734, #72264503
maximal normal subgroup of	#72240727, #72240734, #72264503

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299742);