#493532

order	512
nilpotency class	4
isoclinism	isoclinic class of #33531
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3584	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3156 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240843, #72249908, #72249912, …
maximal normal subgroup of	#72240843, #72249908, #72249912, …
maximal quotient of	#72264995, #72265396, #72307640

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400728);