#354288

order	512
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1870 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303578, #72303580
maximal normal subgroup of	#72303578, #72303580

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261484);