#372912

order	512
nilpotency class	3
isoclinism	isoclinic class of #24986
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2572
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

903 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)(3,4)(5,6,10,15,11,16,9,14)(7,17,21,29,19,28,20,12)(8,22,25,32,26,31,24,13)(18,33,23,36,34,27,35,30);(2,4)(5,7,11,19)(6,12)(8,23)(9,20,10,21)(13,30,32,36)(14,28)(15,17)(16,29)(18,24)(22,33,31,27)(25,34)(26,35);(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,27)(14,31)(15,22)(16,32)(17,30)(19,34)(20,35)(21,23)(28,36)(29,33);(2,4)(6,14,16,15)(7,19)(12,17,29,28)(13,31,32,22)(18,34)(20,21)(23,35)(27,30,33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280664, #72280718
maximal normal subgroup of	#72280664, #72280718

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280108);