#374128

order	512
nilpotency class	3
isoclinism	isoclinic class of #24472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2417 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)(5,6,12,18)(7,19,22,32)(8,23,27,34)(9,28,30,31)(10,17,11,16)(13,25,33,21)(14,26,35,20)(15,24,36,29);(5,7)(6,13)(8,24)(9,20)(10,25)(11,21)(12,22)(14,15)(16,32)(17,19)(18,33)(23,28)(26,30)(27,29)(31,34)(35,36);(2,4)(5,8,10,26,12,27,11,20)(6,14,16,34,18,35,17,23)(7,9,25,24,22,30,21,29)(13,31,32,36,33,28,19,15);(5,9)(6,15)(7,20)(8,25)(10,24)(11,29)(12,30)(13,14)(16,31)(17,28)(18,36)(19,23)(21,27)(22,26)(32,34)(33,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281324);