#493772

order	512
nilpotency class	4
isoclinism	isoclinic class of #11663
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286679, #72286839, D₄×#2145, …
maximal normal subgroup of	#72286679, #72286839, D₄×#2145, …
maximal quotient of	#72310126

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400968);