#493788

order	512
nilpotency class	4
isoclinism	isoclinic class of #2230
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#476
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#476
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1100 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)(7,19)(8,23,25,36)(9,15)(10,26)(11,16)(12,18)(13,21)(14,20,31,35)(17,22)(24,32,28,33)(27,29,34,30);(1,2,3,4)(5,7,9,21,12,22,11,10)(6,13,16,19,18,26,15,17)(8,20,24,34,25,35,28,27)(14,23,30,33,31,36,29,32);(5,8)(6,14)(7,20)(9,24)(10,27)(11,28)(12,25)(13,23)(15,29)(16,30)(17,32)(18,31)(19,33)(21,34)(22,35)(26,36);(5,9,12,11)(6,15,18,16)(7,21,22,10)(8,24,25,28)(13,17,26,19)(14,29,31,30)(20,34,35,27)(23,32,36,33)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=i,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,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	#72224792, #72224799, #72224801, …
maximal normal subgroup of	#72224792, #72224799, #72224801, …
maximal quotient of	C₄×#12823

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400984);